Elements of graded rings of modular forms for Hecke triangle groups¶

AUTHORS:

Jonas Jermann (2013): initial version

class sage.modular.modform_hecketriangle.graded_ring_element.FormsRingElement(parent, rat)[source]¶

Bases: CommutativeAlgebraElement, UniqueRepresentation

Element of a FormsRing.

AT = Analytic Type[source]¶

AnalyticType[source]¶: alias of AnalyticType

analytic_type()[source]¶

Return the analytic type of self.

EXAMPLES:

Sage

sage: from sage.modular.modform_hecketriangle.graded_ring import QuasiMeromorphicModularFormsRing
sage: from sage.modular.modform_hecketriangle.space import QuasiMeromorphicModularForms

sage: # needs sage.symbolic
sage: x, y, z, d = var("x,y,z,d")
sage: QuasiMeromorphicModularFormsRing(n=5)(x/z+d).analytic_type()
quasi meromorphic modular
sage: QuasiMeromorphicModularFormsRing(n=5)((y^3-z^5)/(x^5-y^2)+y-d).analytic_type()
quasi weakly holomorphic modular
sage: QuasiMeromorphicModularFormsRing(n=5)(x^2+y-d).analytic_type()
modular

sage: QuasiMeromorphicModularForms(n=18).J_inv().analytic_type()
weakly holomorphic modular
sage: QuasiMeromorphicModularForms(n=18).f_inf().analytic_type()
cuspidal
sage: QuasiMeromorphicModularForms(n=infinity).f_inf().analytic_type()
modular

Python

>>> from sage.all import *
>>> from sage.modular.modform_hecketriangle.graded_ring import QuasiMeromorphicModularFormsRing
>>> from sage.modular.modform_hecketriangle.space import QuasiMeromorphicModularForms

>>> # needs sage.symbolic
>>> x, y, z, d = var("x,y,z,d")
>>> QuasiMeromorphicModularFormsRing(n=Integer(5))(x/z+d).analytic_type()
quasi meromorphic modular
>>> QuasiMeromorphicModularFormsRing(n=Integer(5))((y**Integer(3)-z**Integer(5))/(x**Integer(5)-y**Integer(2))+y-d).analytic_type()
quasi weakly holomorphic modular
>>> QuasiMeromorphicModularFormsRing(n=Integer(5))(x**Integer(2)+y-d).analytic_type()
modular

>>> QuasiMeromorphicModularForms(n=Integer(18)).J_inv().analytic_type()
weakly holomorphic modular
>>> QuasiMeromorphicModularForms(n=Integer(18)).f_inf().analytic_type()
cuspidal
>>> QuasiMeromorphicModularForms(n=infinity).f_inf().analytic_type()
modular

Sage Live

from sage.modular.modform_hecketriangle.graded_ring import QuasiMeromorphicModularFormsRing
from sage.modular.modform_hecketriangle.space import QuasiMeromorphicModularForms
# needs sage.symbolic
x, y, z, d = var("x,y,z,d")
QuasiMeromorphicModularFormsRing(n=5)(x/z+d).analytic_type()
QuasiMeromorphicModularFormsRing(n=5)((y^3-z^5)/(x^5-y^2)+y-d).analytic_type()
QuasiMeromorphicModularFormsRing(n=5)(x^2+y-d).analytic_type()
QuasiMeromorphicModularForms(n=18).J_inv().analytic_type()
QuasiMeromorphicModularForms(n=18).f_inf().analytic_type()
QuasiMeromorphicModularForms(n=infinity).f_inf().analytic_type()

as_ring_element()[source]¶

Coerce self into the graded ring of its parent.

EXAMPLES:

Sage

sage: from sage.modular.modform_hecketriangle.space import CuspForms
sage: Delta = CuspForms(k=12).Delta()
sage: Delta.parent()
CuspForms(n=3, k=12, ep=1) over Integer Ring
sage: Delta.as_ring_element()
f_rho^3*d - f_i^2*d
sage: Delta.as_ring_element().parent()
CuspFormsRing(n=3) over Integer Ring

sage: CuspForms(n=infinity, k=12).Delta().as_ring_element()
-E4^2*f_i^2*d + E4^3*d

Python

>>> from sage.all import *
>>> from sage.modular.modform_hecketriangle.space import CuspForms
>>> Delta = CuspForms(k=Integer(12)).Delta()
>>> Delta.parent()
CuspForms(n=3, k=12, ep=1) over Integer Ring
>>> Delta.as_ring_element()
f_rho^3*d - f_i^2*d
>>> Delta.as_ring_element().parent()
CuspFormsRing(n=3) over Integer Ring

>>> CuspForms(n=infinity, k=Integer(12)).Delta().as_ring_element()
-E4^2*f_i^2*d + E4^3*d

Sage Live

from sage.modular.modform_hecketriangle.space import CuspForms
Delta = CuspForms(k=12).Delta()
Delta.parent()
Delta.as_ring_element()
Delta.as_ring_element().parent()
CuspForms(n=infinity, k=12).Delta().as_ring_element()

base_ring()[source]¶

Return base ring of self.parent().

EXAMPLES:

Sage

sage: from sage.modular.modform_hecketriangle.space import ModularForms
sage: ModularForms(n=12, k=4, base_ring=CC).E4().base_ring()
Complex Field with 53 bits of precision

Python

>>> from sage.all import *
>>> from sage.modular.modform_hecketriangle.space import ModularForms
>>> ModularForms(n=Integer(12), k=Integer(4), base_ring=CC).E4().base_ring()
Complex Field with 53 bits of precision

Sage Live

from sage.modular.modform_hecketriangle.space import ModularForms
ModularForms(n=12, k=4, base_ring=CC).E4().base_ring()

coeff_ring()[source]¶

Return coefficient ring of self.

EXAMPLES:

Sage

sage: from sage.modular.modform_hecketriangle.graded_ring import ModularFormsRing
sage: ModularFormsRing().E6().coeff_ring()
Fraction Field of Univariate Polynomial Ring in d over Integer Ring

Python

>>> from sage.all import *
>>> from sage.modular.modform_hecketriangle.graded_ring import ModularFormsRing
>>> ModularFormsRing().E6().coeff_ring()
Fraction Field of Univariate Polynomial Ring in d over Integer Ring

Sage Live

from sage.modular.modform_hecketriangle.graded_ring import ModularFormsRing
ModularFormsRing().E6().coeff_ring()

degree()[source]¶

Return the degree of self in the graded ring. If self is not homogeneous, then (None, None) is returned.

EXAMPLES:

Sage

sage: from sage.modular.modform_hecketriangle.graded_ring import QuasiModularFormsRing
sage: from sage.modular.modform_hecketriangle.space import ModularForms
sage: x, y, z, d = var("x,y,z,d")                                           # needs sage.symbolic
sage: QuasiModularFormsRing()(x+y).degree() == (None, None)                 # needs sage.symbolic
True
sage: ModularForms(n=18).f_i().degree()
(9/4, -1)
sage: ModularForms(n=infinity).f_rho().degree()
(0, 1)

Python

>>> from sage.all import *
>>> from sage.modular.modform_hecketriangle.graded_ring import QuasiModularFormsRing
>>> from sage.modular.modform_hecketriangle.space import ModularForms
>>> x, y, z, d = var("x,y,z,d")                                           # needs sage.symbolic
>>> QuasiModularFormsRing()(x+y).degree() == (None, None)                 # needs sage.symbolic
True
>>> ModularForms(n=Integer(18)).f_i().degree()
(9/4, -1)
>>> ModularForms(n=infinity).f_rho().degree()
(0, 1)

Sage Live

from sage.modular.modform_hecketriangle.graded_ring import QuasiModularFormsRing
from sage.modular.modform_hecketriangle.space import ModularForms
x, y, z, d = var("x,y,z,d")                                           # needs sage.symbolic
QuasiModularFormsRing()(x+y).degree() == (None, None)                 # needs sage.symbolic
ModularForms(n=18).f_i().degree()
ModularForms(n=infinity).f_rho().degree()

denominator()[source]¶

Return the denominator of self. I.e. the (properly reduced) new form corresponding to the numerator of self.rat().

Note that the parent of self might (probably will) change.

EXAMPLES:

Sage

sage: # needs sage.symbolic
sage: from sage.modular.modform_hecketriangle.graded_ring import QuasiMeromorphicModularFormsRing
sage: from sage.modular.modform_hecketriangle.space import QuasiMeromorphicModularForms
sage: x, y, z, d = var("x,y,z,d")
sage: QuasiMeromorphicModularFormsRing(n=5).Delta().full_reduce().denominator()
1 + O(q^5)
sage: QuasiMeromorphicModularFormsRing(n=5)((y^3-z^5)/(x^5-y^2)+y-d).denominator()
f_rho^5 - f_i^2
sage: QuasiMeromorphicModularFormsRing(n=5)((y^3-z^5)/(x^5-y^2)+y-d).denominator().parent()
QuasiModularFormsRing(n=5) over Integer Ring
sage: QuasiMeromorphicModularForms(n=5, k=-2, ep=-1)(x/y).denominator()
1 - 13/(40*d)*q - 351/(64000*d^2)*q^2 - 13819/(76800000*d^3)*q^3 - 1163669/(491520000000*d^4)*q^4 + O(q^5)
sage: QuasiMeromorphicModularForms(n=5, k=-2, ep=-1)(x/y).denominator().parent()
QuasiModularForms(n=5, k=10/3, ep=-1) over Integer Ring
sage: (QuasiMeromorphicModularForms(n=infinity, k=-6, ep=-1)(y/(x*(x-y^2)))).denominator()
-64*q - 512*q^2 - 768*q^3 + 4096*q^4 + O(q^5)
sage: (QuasiMeromorphicModularForms(n=infinity, k=-6, ep=-1)(y/(x*(x-y^2)))).denominator().parent()
QuasiModularForms(n=+Infinity, k=8, ep=1) over Integer Ring

Python

>>> from sage.all import *
>>> # needs sage.symbolic
>>> from sage.modular.modform_hecketriangle.graded_ring import QuasiMeromorphicModularFormsRing
>>> from sage.modular.modform_hecketriangle.space import QuasiMeromorphicModularForms
>>> x, y, z, d = var("x,y,z,d")
>>> QuasiMeromorphicModularFormsRing(n=Integer(5)).Delta().full_reduce().denominator()
1 + O(q^5)
>>> QuasiMeromorphicModularFormsRing(n=Integer(5))((y**Integer(3)-z**Integer(5))/(x**Integer(5)-y**Integer(2))+y-d).denominator()
f_rho^5 - f_i^2
>>> QuasiMeromorphicModularFormsRing(n=Integer(5))((y**Integer(3)-z**Integer(5))/(x**Integer(5)-y**Integer(2))+y-d).denominator().parent()
QuasiModularFormsRing(n=5) over Integer Ring
>>> QuasiMeromorphicModularForms(n=Integer(5), k=-Integer(2), ep=-Integer(1))(x/y).denominator()
1 - 13/(40*d)*q - 351/(64000*d^2)*q^2 - 13819/(76800000*d^3)*q^3 - 1163669/(491520000000*d^4)*q^4 + O(q^5)
>>> QuasiMeromorphicModularForms(n=Integer(5), k=-Integer(2), ep=-Integer(1))(x/y).denominator().parent()
QuasiModularForms(n=5, k=10/3, ep=-1) over Integer Ring
>>> (QuasiMeromorphicModularForms(n=infinity, k=-Integer(6), ep=-Integer(1))(y/(x*(x-y**Integer(2))))).denominator()
-64*q - 512*q^2 - 768*q^3 + 4096*q^4 + O(q^5)
>>> (QuasiMeromorphicModularForms(n=infinity, k=-Integer(6), ep=-Integer(1))(y/(x*(x-y**Integer(2))))).denominator().parent()
QuasiModularForms(n=+Infinity, k=8, ep=1) over Integer Ring

Sage Live

# needs sage.symbolic
from sage.modular.modform_hecketriangle.graded_ring import QuasiMeromorphicModularFormsRing
from sage.modular.modform_hecketriangle.space import QuasiMeromorphicModularForms
x, y, z, d = var("x,y,z,d")
QuasiMeromorphicModularFormsRing(n=5).Delta().full_reduce().denominator()
QuasiMeromorphicModularFormsRing(n=5)((y^3-z^5)/(x^5-y^2)+y-d).denominator()
QuasiMeromorphicModularFormsRing(n=5)((y^3-z^5)/(x^5-y^2)+y-d).denominator().parent()
QuasiMeromorphicModularForms(n=5, k=-2, ep=-1)(x/y).denominator()
QuasiMeromorphicModularForms(n=5, k=-2, ep=-1)(x/y).denominator().parent()
(QuasiMeromorphicModularForms(n=infinity, k=-6, ep=-1)(y/(x*(x-y^2)))).denominator()
(QuasiMeromorphicModularForms(n=infinity, k=-6, ep=-1)(y/(x*(x-y^2)))).denominator().parent()

derivative()[source]¶

Return the derivative d/dq = lambda/(2*pi*i) d/dtau of self.

Note that the parent might (probably will) change. In particular its analytic type will be extended to contain “quasi”.

If parent.has_reduce_hom() == True then the result is reduced to be an element of the corresponding forms space if possible.

In particular this is the case if self is a (homogeneous) element of a forms space.

EXAMPLES:

Sage

sage: from sage.modular.modform_hecketriangle.graded_ring import QuasiMeromorphicModularFormsRing
sage: MR = QuasiMeromorphicModularFormsRing(n=7, red_hom=True)
sage: n = MR.hecke_n()
sage: E2 = MR.E2().full_reduce()
sage: E6 = MR.E6().full_reduce()
sage: f_rho = MR.f_rho().full_reduce()
sage: f_i = MR.f_i().full_reduce()
sage: f_inf = MR.f_inf().full_reduce()

sage: derivative(f_rho) == 1/n * (f_rho*E2 - f_i)
True
sage: derivative(f_i)   == 1/2 * (f_i*E2 - f_rho**(n-1))
True
sage: derivative(f_inf) == f_inf * E2
True
sage: derivative(f_inf).parent()
QuasiCuspForms(n=7, k=38/5, ep=-1) over Integer Ring
sage: derivative(E2)    == (n-2)/(4*n) * (E2**2 - f_rho**(n-2))
True
sage: derivative(E2).parent()
QuasiModularForms(n=7, k=4, ep=1) over Integer Ring

sage: MR = QuasiMeromorphicModularFormsRing(n=infinity, red_hom=True)
sage: E2 = MR.E2().full_reduce()
sage: E4 = MR.E4().full_reduce()
sage: E6 = MR.E6().full_reduce()
sage: f_i = MR.f_i().full_reduce()
sage: f_inf = MR.f_inf().full_reduce()

sage: derivative(E4)    == E4 * (E2 - f_i)
True
sage: derivative(f_i)   == 1/2 * (f_i*E2 - E4)
True
sage: derivative(f_inf) == f_inf * E2
True
sage: derivative(f_inf).parent()
QuasiModularForms(n=+Infinity, k=6, ep=-1) over Integer Ring
sage: derivative(E2)    == 1/4 * (E2**2 - E4)
True
sage: derivative(E2).parent()
QuasiModularForms(n=+Infinity, k=4, ep=1) over Integer Ring

Python

>>> from sage.all import *
>>> from sage.modular.modform_hecketriangle.graded_ring import QuasiMeromorphicModularFormsRing
>>> MR = QuasiMeromorphicModularFormsRing(n=Integer(7), red_hom=True)
>>> n = MR.hecke_n()
>>> E2 = MR.E2().full_reduce()
>>> E6 = MR.E6().full_reduce()
>>> f_rho = MR.f_rho().full_reduce()
>>> f_i = MR.f_i().full_reduce()
>>> f_inf = MR.f_inf().full_reduce()

>>> derivative(f_rho) == Integer(1)/n * (f_rho*E2 - f_i)
True
>>> derivative(f_i)   == Integer(1)/Integer(2) * (f_i*E2 - f_rho**(n-Integer(1)))
True
>>> derivative(f_inf) == f_inf * E2
True
>>> derivative(f_inf).parent()
QuasiCuspForms(n=7, k=38/5, ep=-1) over Integer Ring
>>> derivative(E2)    == (n-Integer(2))/(Integer(4)*n) * (E2**Integer(2) - f_rho**(n-Integer(2)))
True
>>> derivative(E2).parent()
QuasiModularForms(n=7, k=4, ep=1) over Integer Ring

>>> MR = QuasiMeromorphicModularFormsRing(n=infinity, red_hom=True)
>>> E2 = MR.E2().full_reduce()
>>> E4 = MR.E4().full_reduce()
>>> E6 = MR.E6().full_reduce()
>>> f_i = MR.f_i().full_reduce()
>>> f_inf = MR.f_inf().full_reduce()

>>> derivative(E4)    == E4 * (E2 - f_i)
True
>>> derivative(f_i)   == Integer(1)/Integer(2) * (f_i*E2 - E4)
True
>>> derivative(f_inf) == f_inf * E2
True
>>> derivative(f_inf).parent()
QuasiModularForms(n=+Infinity, k=6, ep=-1) over Integer Ring
>>> derivative(E2)    == Integer(1)/Integer(4) * (E2**Integer(2) - E4)
True
>>> derivative(E2).parent()
QuasiModularForms(n=+Infinity, k=4, ep=1) over Integer Ring

Sage Live

from sage.modular.modform_hecketriangle.graded_ring import QuasiMeromorphicModularFormsRing
MR = QuasiMeromorphicModularFormsRing(n=7, red_hom=True)
n = MR.hecke_n()
E2 = MR.E2().full_reduce()
E6 = MR.E6().full_reduce()
f_rho = MR.f_rho().full_reduce()
f_i = MR.f_i().full_reduce()
f_inf = MR.f_inf().full_reduce()
derivative(f_rho) == 1/n * (f_rho*E2 - f_i)
derivative(f_i)   == 1/2 * (f_i*E2 - f_rho**(n-1))
derivative(f_inf) == f_inf * E2
derivative(f_inf).parent()
derivative(E2)    == (n-2)/(4*n) * (E2**2 - f_rho**(n-2))
derivative(E2).parent()
MR = QuasiMeromorphicModularFormsRing(n=infinity, red_hom=True)
E2 = MR.E2().full_reduce()
E4 = MR.E4().full_reduce()
E6 = MR.E6().full_reduce()
f_i = MR.f_i().full_reduce()
f_inf = MR.f_inf().full_reduce()
derivative(E4)    == E4 * (E2 - f_i)
derivative(f_i)   == 1/2 * (f_i*E2 - E4)
derivative(f_inf) == f_inf * E2
derivative(f_inf).parent()
derivative(E2)    == 1/4 * (E2**2 - E4)
derivative(E2).parent()

diff_op(op, new_parent=None)[source]¶

Return the differential operator op applied to self. If parent.has_reduce_hom() == True then the result is reduced to be an element of the corresponding forms space if possible.

INPUT:

op – an element of self.parent().diff_alg(). I.e. an element of the algebra over QQ of differential operators generated by X, Y, Z, dX, dY, DZ, where e.g. X corresponds to the multiplication by x (resp. f_rho) and dX corresponds to d/dx.

To expect a homogeneous result after applying the operator to a homogeneous element it should should be homogeneous operator (with respect to the usual, special grading).
new_parent – try to convert the result to the specified new_parent. If new_parent == None (default) then the parent is extended to a “quasi meromorphic” ring.

OUTPUT: the new element

EXAMPLES:

Sage

sage: from sage.modular.modform_hecketriangle.graded_ring import QuasiMeromorphicModularFormsRing
sage: MR = QuasiMeromorphicModularFormsRing(n=8, red_hom=True)
sage: (X,Y,Z,dX,dY,dZ) = MR.diff_alg().gens()
sage: n = MR.hecke_n()
sage: mul_op = 4/(n-2)*X*dX + 2*n/(n-2)*Y*dY + 2*Z*dZ
sage: der_op = MR._derivative_op()
sage: ser_op = MR._serre_derivative_op()
sage: der_op == ser_op + (n-2)/(4*n)*Z*mul_op
True

sage: Delta = MR.Delta().full_reduce()
sage: E2 = MR.E2().full_reduce()
sage: Delta.diff_op(mul_op) == 12*Delta
True
sage: Delta.diff_op(mul_op).parent()
QuasiMeromorphicModularForms(n=8, k=12, ep=1) over Integer Ring
sage: Delta.diff_op(mul_op, Delta.parent()).parent()
CuspForms(n=8, k=12, ep=1) over Integer Ring
sage: E2.diff_op(mul_op, E2.parent()) == 2*E2
True
sage: Delta.diff_op(Z*mul_op, Delta.parent().extend_type("quasi", ring=True)) == 12*E2*Delta
True

sage: ran_op = X + Y*X*dY*dX + dZ + dX^2
sage: Delta.diff_op(ran_op)
f_rho^19*d + 306*f_rho^16*d - f_rho^11*f_i^2*d - 20*f_rho^10*f_i^2*d - 90*f_rho^8*f_i^2*d
sage: E2.diff_op(ran_op)
f_rho*E2 + 1

sage: MR = QuasiMeromorphicModularFormsRing(n=infinity, red_hom=True)
sage: (X,Y,Z,dX,dY,dZ) = MR.diff_alg().gens()
sage: mul_op = 4*X*dX + 2*Y*dY + 2*Z*dZ
sage: der_op = MR._derivative_op()
sage: ser_op = MR._serre_derivative_op()
sage: der_op == ser_op + Z/4*mul_op
True

sage: Delta = MR.Delta().full_reduce()
sage: E2 = MR.E2().full_reduce()
sage: Delta.diff_op(mul_op) == 12*Delta
True
sage: Delta.diff_op(mul_op).parent()
QuasiMeromorphicModularForms(n=+Infinity, k=12, ep=1) over Integer Ring
sage: Delta.diff_op(mul_op, Delta.parent()).parent()
CuspForms(n=+Infinity, k=12, ep=1) over Integer Ring
sage: E2.diff_op(mul_op, E2.parent()) == 2*E2
True
sage: Delta.diff_op(Z*mul_op, Delta.parent().extend_type("quasi", ring=True)) == 12*E2*Delta
True

sage: ran_op = X + Y*X*dY*dX + dZ + dX^2
sage: Delta.diff_op(ran_op)
-E4^3*f_i^2*d + E4^4*d - 4*E4^2*f_i^2*d - 2*f_i^2*d + 6*E4*d
sage: E2.diff_op(ran_op)
E4*E2 + 1

Python

>>> from sage.all import *
>>> from sage.modular.modform_hecketriangle.graded_ring import QuasiMeromorphicModularFormsRing
>>> MR = QuasiMeromorphicModularFormsRing(n=Integer(8), red_hom=True)
>>> (X,Y,Z,dX,dY,dZ) = MR.diff_alg().gens()
>>> n = MR.hecke_n()
>>> mul_op = Integer(4)/(n-Integer(2))*X*dX + Integer(2)*n/(n-Integer(2))*Y*dY + Integer(2)*Z*dZ
>>> der_op = MR._derivative_op()
>>> ser_op = MR._serre_derivative_op()
>>> der_op == ser_op + (n-Integer(2))/(Integer(4)*n)*Z*mul_op
True

>>> Delta = MR.Delta().full_reduce()
>>> E2 = MR.E2().full_reduce()
>>> Delta.diff_op(mul_op) == Integer(12)*Delta
True
>>> Delta.diff_op(mul_op).parent()
QuasiMeromorphicModularForms(n=8, k=12, ep=1) over Integer Ring
>>> Delta.diff_op(mul_op, Delta.parent()).parent()
CuspForms(n=8, k=12, ep=1) over Integer Ring
>>> E2.diff_op(mul_op, E2.parent()) == Integer(2)*E2
True
>>> Delta.diff_op(Z*mul_op, Delta.parent().extend_type("quasi", ring=True)) == Integer(12)*E2*Delta
True

>>> ran_op = X + Y*X*dY*dX + dZ + dX**Integer(2)
>>> Delta.diff_op(ran_op)
f_rho^19*d + 306*f_rho^16*d - f_rho^11*f_i^2*d - 20*f_rho^10*f_i^2*d - 90*f_rho^8*f_i^2*d
>>> E2.diff_op(ran_op)
f_rho*E2 + 1

>>> MR = QuasiMeromorphicModularFormsRing(n=infinity, red_hom=True)
>>> (X,Y,Z,dX,dY,dZ) = MR.diff_alg().gens()
>>> mul_op = Integer(4)*X*dX + Integer(2)*Y*dY + Integer(2)*Z*dZ
>>> der_op = MR._derivative_op()
>>> ser_op = MR._serre_derivative_op()
>>> der_op == ser_op + Z/Integer(4)*mul_op
True

>>> Delta = MR.Delta().full_reduce()
>>> E2 = MR.E2().full_reduce()
>>> Delta.diff_op(mul_op) == Integer(12)*Delta
True
>>> Delta.diff_op(mul_op).parent()
QuasiMeromorphicModularForms(n=+Infinity, k=12, ep=1) over Integer Ring
>>> Delta.diff_op(mul_op, Delta.parent()).parent()
CuspForms(n=+Infinity, k=12, ep=1) over Integer Ring
>>> E2.diff_op(mul_op, E2.parent()) == Integer(2)*E2
True
>>> Delta.diff_op(Z*mul_op, Delta.parent().extend_type("quasi", ring=True)) == Integer(12)*E2*Delta
True

>>> ran_op = X + Y*X*dY*dX + dZ + dX**Integer(2)
>>> Delta.diff_op(ran_op)
-E4^3*f_i^2*d + E4^4*d - 4*E4^2*f_i^2*d - 2*f_i^2*d + 6*E4*d
>>> E2.diff_op(ran_op)
E4*E2 + 1

Sage Live

from sage.modular.modform_hecketriangle.graded_ring import QuasiMeromorphicModularFormsRing
MR = QuasiMeromorphicModularFormsRing(n=8, red_hom=True)
(X,Y,Z,dX,dY,dZ) = MR.diff_alg().gens()
n = MR.hecke_n()
mul_op = 4/(n-2)*X*dX + 2*n/(n-2)*Y*dY + 2*Z*dZ
der_op = MR._derivative_op()
ser_op = MR._serre_derivative_op()
der_op == ser_op + (n-2)/(4*n)*Z*mul_op
Delta = MR.Delta().full_reduce()
E2 = MR.E2().full_reduce()
Delta.diff_op(mul_op) == 12*Delta
Delta.diff_op(mul_op).parent()
Delta.diff_op(mul_op, Delta.parent()).parent()
E2.diff_op(mul_op, E2.parent()) == 2*E2
Delta.diff_op(Z*mul_op, Delta.parent().extend_type("quasi", ring=True)) == 12*E2*Delta
ran_op = X + Y*X*dY*dX + dZ + dX^2
Delta.diff_op(ran_op)
E2.diff_op(ran_op)
MR = QuasiMeromorphicModularFormsRing(n=infinity, red_hom=True)
(X,Y,Z,dX,dY,dZ) = MR.diff_alg().gens()
mul_op = 4*X*dX + 2*Y*dY + 2*Z*dZ
der_op = MR._derivative_op()
ser_op = MR._serre_derivative_op()
der_op == ser_op + Z/4*mul_op
Delta = MR.Delta().full_reduce()
E2 = MR.E2().full_reduce()
Delta.diff_op(mul_op) == 12*Delta
Delta.diff_op(mul_op).parent()
Delta.diff_op(mul_op, Delta.parent()).parent()
E2.diff_op(mul_op, E2.parent()) == 2*E2
Delta.diff_op(Z*mul_op, Delta.parent().extend_type("quasi", ring=True)) == 12*E2*Delta
ran_op = X + Y*X*dY*dX + dZ + dX^2
Delta.diff_op(ran_op)
E2.diff_op(ran_op)

ep()[source]¶

Return the multiplier of self.

EXAMPLES:

Sage

sage: from sage.modular.modform_hecketriangle.graded_ring import QuasiModularFormsRing
sage: from sage.modular.modform_hecketriangle.space import ModularForms
sage: x, y, z, d = var("x,y,z,d")                                           # needs sage.symbolic
sage: QuasiModularFormsRing()(x+y).ep() is None                             # needs sage.symbolic
True
sage: ModularForms(n=18).f_i().ep()
-1
sage: ModularForms(n=infinity).E2().ep()
-1

Python

>>> from sage.all import *
>>> from sage.modular.modform_hecketriangle.graded_ring import QuasiModularFormsRing
>>> from sage.modular.modform_hecketriangle.space import ModularForms
>>> x, y, z, d = var("x,y,z,d")                                           # needs sage.symbolic
>>> QuasiModularFormsRing()(x+y).ep() is None                             # needs sage.symbolic
True
>>> ModularForms(n=Integer(18)).f_i().ep()
-1
>>> ModularForms(n=infinity).E2().ep()
-1

Sage Live

from sage.modular.modform_hecketriangle.graded_ring import QuasiModularFormsRing
from sage.modular.modform_hecketriangle.space import ModularForms
x, y, z, d = var("x,y,z,d")                                           # needs sage.symbolic
QuasiModularFormsRing()(x+y).ep() is None                             # needs sage.symbolic
ModularForms(n=18).f_i().ep()
ModularForms(n=infinity).E2().ep()

evaluate(tau, prec=None, num_prec=None, check=False)[source]¶

Try to return self evaluated at a point tau in the upper half plane, where self is interpreted as a function in tau, where q=exp(2*pi*i*tau).

Note that this interpretation might not make sense (and fail) for certain (many) choices of (base_ring, tau.parent()).

It is possible to evaluate at points of HyperbolicPlane. In this case the coordinates of the upper half plane model are used.

To obtain a precise and fast result the parameters prec and num_prec both have to be considered/balanced. A high prec value is usually quite costly.

INPUT:

tau – infinity or an element of the upper half plane. E.g. with parent AA or CC.
prec – integer, namely the precision used for the Fourier expansion. If prec == None (default) then the default precision of self.parent() is used
num_prec – integer, namely the minimal numerical precision used for tau and d. If num_prec == None (default) then the default numerical precision of self.parent() is used.
check – if True then the order of tau is checked. Otherwise the order is only considered for tau = infinity, i, rho, -1/rho. Default: False.

OUTPUT:

The (numerical) evaluated function value.

ALGORITHM:

If the order of self at tau is known and nonzero: Return 0 resp. infinity.
Else if tau==infinity and the order is zero: Return the constant Fourier coefficient of self.
Else if self is homogeneous and modular:
1. Because of the (modular) transformation property of self the evaluation at tau is given by the evaluation at w multiplied by aut_factor(A,w).
2. The evaluation at w is calculated by evaluating the truncated Fourier expansion of self at q(w).
Note that this is much faster and more precise than a direct evaluation at tau.
Else if self is exactly E2:
1. The same procedure as before is applied (with the aut_factor from the corresponding modular space).
2. Except that at the end a correction term for the quasimodular form E2 of the form 4*lambda/(2*pi*i)*n/(n-2) * c*(c*w + d) (resp. 4/(pi*i) * c*(c*w + d) for n=infinity) has to be added, where lambda = 2*cos(pi/n) (resp lambda = 2 for n=infinity) and c,d are the lower entries of the matrix A.
Else:
1. Evaluate f_rho, f_i, E2 at tau using the above procedures. If n=infinity use E4 instead of f_rho.
2. Substitute x=f_rho(tau), y=f_i(tau), z=E2(tau) and the numerical value of d for d in self.rat(). If n=infinity then substitute x=E4(tau) instead.

EXAMPLES:

Sage

sage: from sage.modular.modform_hecketriangle.hecke_triangle_groups import HeckeTriangleGroup
sage: from sage.modular.modform_hecketriangle.graded_ring import QuasiMeromorphicModularFormsRing
sage: MR = QuasiMeromorphicModularFormsRing(n=5, red_hom=True)
sage: f_rho = MR.f_rho().full_reduce()
sage: f_i   = MR.f_i().full_reduce()
sage: f_inf = MR.f_inf().full_reduce()
sage: E2    = MR.E2().full_reduce()
sage: E4    = MR.E4().full_reduce()
sage: rho   = MR.group().rho()

sage: f_rho(rho)
0
sage: f_rho(rho + 1e-100)    # since rho == rho + 1e-100
0
sage: f_rho(rho + 1e-6)
2.525...e-10 - 3.884...e-6*I
sage: f_i(i)
0
sage: f_i(i + 1e-1000)  # rel tol 5e-2
-6.08402217494586e-14 - 4.10147008296517e-1000*I
sage: f_inf(infinity)
0

sage: i = I = QuadraticField(-1, 'I').gen()
sage: z = -1/(-1/(2*i+30)-1)
sage: z
2/965*I + 934/965
sage: E4(z)
32288.05588811... - 118329.8566016...*I
sage: E4(z, prec=30, num_prec=100)    # long time
32288.0558872351130041311053... - 118329.856600349999751420381...*I
sage: E2(z)
409.3144737105... + 100.6926857489...*I
sage: E2(z, prec=30, num_prec=100)    # long time
409.314473710489761254584951... + 100.692685748952440684513866...*I
sage: (E2^2-E4)(z)
125111.2655383... + 200759.8039479...*I
sage: (E2^2-E4)(z, prec=30, num_prec=100)    # long time
125111.265538336196262200469... + 200759.803948009905410385699...*I

sage: (E2^2-E4)(infinity)
0
sage: (1/(E2^2-E4))(infinity)
+Infinity
sage: ((E2^2-E4)/f_inf)(infinity)
-3/(10*d)

sage: G = HeckeTriangleGroup(n=8)
sage: MR = QuasiMeromorphicModularFormsRing(group=G, red_hom=True)
sage: f_rho = MR.f_rho().full_reduce()
sage: f_i   = MR.f_i().full_reduce()
sage: E2    = MR.E2().full_reduce()

sage: z = AlgebraicField()(1/10+13/10*I)
sage: A = G.V(4)
sage: S = G.S()
sage: T = G.T()
sage: A == (T*S)**3*T
True
sage: az = A.acton(z)
sage: az == (A[0,0]*z + A[0,1]) / (A[1,0]*z + A[1,1])
True

sage: f_rho(z)
1.03740476727... + 0.0131941034523...*I
sage: f_rho(az)
-2.29216470688... - 1.46235057536...*I
sage: k = f_rho.weight()
sage: aut_fact = f_rho.ep()^3 * (((T*S)**2*T).acton(z)/AlgebraicField()(i))**k * (((T*S)*T).acton(z)/AlgebraicField()(i))**k * (T.acton(z)/AlgebraicField()(i))**k
sage: abs(aut_fact - f_rho.parent().aut_factor(A, z)) < 1e-12
True
sage: aut_fact * f_rho(z)
-2.29216470688... - 1.46235057536...*I

sage: f_rho.parent().default_num_prec(1000)
sage: f_rho.parent().default_prec(300)
sage: (f_rho.q_expansion_fixed_d().polynomial())(exp((2*pi*i).n(1000)*z/G.lam()))    # long time
1.0374047672719462149821251... + 0.013194103452368974597290332...*I
sage: (f_rho.q_expansion_fixed_d().polynomial())(exp((2*pi*i).n(1000)*az/G.lam()))    # long time
-2.2921647068881834598616367... - 1.4623505753697635207183406...*I

sage: f_i(z)
0.667489320423... - 0.118902824870...*I
sage: f_i(az)
14.5845388476... - 28.4604652892...*I
sage: k = f_i.weight()
sage: aut_fact = f_i.ep()^3 * (((T*S)**2*T).acton(z)/AlgebraicField()(i))**k * (((T*S)*T).acton(z)/AlgebraicField()(i))**k * (T.acton(z)/AlgebraicField()(i))**k
sage: abs(aut_fact - f_i.parent().aut_factor(A, z)) < 1e-12
True
sage: aut_fact * f_i(z)
14.5845388476... - 28.4604652892...*I

sage: f_i.parent().default_num_prec(1000)
sage: f_i.parent().default_prec(300)
sage: (f_i.q_expansion_fixed_d().polynomial())(exp((2*pi*i).n(1000)*z/G.lam()))    # long time
0.66748932042300250077433252... - 0.11890282487028677063054267...*I
sage: (f_i.q_expansion_fixed_d().polynomial())(exp((2*pi*i).n(1000)*az/G.lam()))    # long time
14.584538847698600875918891... - 28.460465289220303834894855...*I

sage: f = f_rho*E2
sage: f(z)
0.966024386418... - 0.0138894699429...*I
sage: f(az)
-15.9978074989... - 29.2775758341...*I
sage: k = f.weight()
sage: aut_fact = f.ep()^3 * (((T*S)**2*T).acton(z)/AlgebraicField()(i))**k * (((T*S)*T).acton(z)/AlgebraicField()(i))**k * (T.acton(z)/AlgebraicField()(i))**k
sage: abs(aut_fact - f.parent().aut_factor(A, z)) < 1e-12
True
sage: k2 = f_rho.weight()
sage: aut_fact2 = f_rho.ep() * (((T*S)**2*T).acton(z)/AlgebraicField()(i))**k2 * (((T*S)*T).acton(z)/AlgebraicField()(i))**k2 * (T.acton(z)/AlgebraicField()(i))**k2
sage: abs(aut_fact2 - f_rho.parent().aut_factor(A, z)) < 1e-12
True
sage: cor_term = (4 * G.n() / (G.n()-2) * A.c() * (A.c()*z+A.d())) / (2*pi*i).n(1000) * G.lam()
sage: aut_fact*f(z) + cor_term*aut_fact2*f_rho(z)
-15.9978074989... - 29.2775758341...*I

sage: f.parent().default_num_prec(1000)
sage: f.parent().default_prec(300)
sage: (f.q_expansion_fixed_d().polynomial())(exp((2*pi*i).n(1000)*z/G.lam()))    # long time
0.96602438641867296777809436... - 0.013889469942995530807311503...*I
sage: (f.q_expansion_fixed_d().polynomial())(exp((2*pi*i).n(1000)*az/G.lam()))    # long time
-15.997807498958825352887040... - 29.277575834123246063432206...*I

sage: MR = QuasiMeromorphicModularFormsRing(n=infinity, red_hom=True)
sage: f_i   = MR.f_i().full_reduce()
sage: f_inf = MR.f_inf().full_reduce()
sage: E2    = MR.E2().full_reduce()
sage: E4    = MR.E4().full_reduce()

sage: f_i(i)
0
sage: f_i(i + 1e-1000)
2.991...e-12 - 3.048...e-1000*I
sage: f_inf(infinity)
0

sage: z = -1/(-1/(2*i+30)-1)
sage: E4(z, prec=15)
804.0722034... + 211.9278206...*I
sage: E4(z, prec=30, num_prec=100)    # long time
803.928382417... + 211.889914044...*I
sage: E2(z)
2.438455612... - 39.48442265...*I
sage: E2(z, prec=30, num_prec=100)    # long time
2.43968197227756036957475... - 39.4842637577742677851431...*I
sage: (E2^2-E4)(z)
-2265.442515... - 380.3197877...*I
sage: (E2^2-E4)(z, prec=30, num_prec=100)    # long time
-2265.44251550679807447320... - 380.319787790548788238792...*I

sage: (E2^2-E4)(infinity)
0
sage: (1/(E2^2-E4))(infinity)
+Infinity
sage: ((E2^2-E4)/f_inf)(infinity)
-1/(2*d)

sage: G = HeckeTriangleGroup(n=Infinity)
sage: z = AlgebraicField()(1/10+13/10*I)
sage: A = G.V(4)
sage: S = G.S()
sage: T = G.T()
sage: A == (T*S)**3*T
True
sage: az = A.acton(z)
sage: az == (A[0,0]*z + A[0,1]) / (A[1,0]*z + A[1,1])
True

sage: f_i(z)
0.6208853409... - 0.1212525492...*I
sage: f_i(az)
6.103314419... + 20.42678597...*I
sage: k = f_i.weight()
sage: aut_fact = f_i.ep()^3 * (((T*S)**2*T).acton(z)/AlgebraicField()(i))**k * (((T*S)*T).acton(z)/AlgebraicField()(i))**k * (T.acton(z)/AlgebraicField()(i))**k
sage: abs(aut_fact - f_i.parent().aut_factor(A, z)) < 1e-12
True
sage: aut_fact * f_i(z)
6.103314419... + 20.42678597...*I

sage: f_i.parent().default_num_prec(1000)
sage: f_i.parent().default_prec(300)
sage: (f_i.q_expansion_fixed_d().polynomial())(exp((2*pi*i).n(1000)*z/G.lam()))    # long time
0.620885340917559158572271... - 0.121252549240996430425967...*I
sage: (f_i.q_expansion_fixed_d().polynomial())(exp((2*pi*i).n(1000)*az/G.lam()))    # long time
6.10331441975198186745017... + 20.4267859728657976382684...*I

sage: f = f_i*E2
sage: f(z)
0.5349190275... - 0.1322370856...*I
sage: f(az)
-140.4711702... + 469.0793692...*I
sage: k = f.weight()
sage: aut_fact = f.ep()^3 * (((T*S)**2*T).acton(z)/AlgebraicField()(i))**k * (((T*S)*T).acton(z)/AlgebraicField()(i))**k * (T.acton(z)/AlgebraicField()(i))**k
sage: abs(aut_fact - f.parent().aut_factor(A, z)) < 1e-12
True
sage: k2 = f_i.weight()
sage: aut_fact2 = f_i.ep() * (((T*S)**2*T).acton(z)/AlgebraicField()(i))**k2 * (((T*S)*T).acton(z)/AlgebraicField()(i))**k2 * (T.acton(z)/AlgebraicField()(i))**k2
sage: abs(aut_fact2 - f_i.parent().aut_factor(A, z)) < 1e-12
True
sage: cor_term = (4 * A.c() * (A.c()*z+A.d())) / (2*pi*i).n(1000) * G.lam()
sage: aut_fact*f(z) + cor_term*aut_fact2*f_i(z)
-140.4711702... + 469.0793692...*I

sage: f.parent().default_num_prec(1000)
sage: f.parent().default_prec(300)
sage: (f.q_expansion_fixed_d().polynomial())(exp((2*pi*i).n(1000)*z/G.lam()))    # long time
0.534919027587592616802582... - 0.132237085641931661668338...*I

sage: (f.q_expansion_fixed_d().polynomial())(exp((2*pi*i).n(1000)*az/G.lam()))    # long time
-140.471170232432551196978... + 469.079369280804086032719...*I

Python

>>> from sage.all import *
>>> from sage.modular.modform_hecketriangle.hecke_triangle_groups import HeckeTriangleGroup
>>> from sage.modular.modform_hecketriangle.graded_ring import QuasiMeromorphicModularFormsRing
>>> MR = QuasiMeromorphicModularFormsRing(n=Integer(5), red_hom=True)
>>> f_rho = MR.f_rho().full_reduce()
>>> f_i   = MR.f_i().full_reduce()
>>> f_inf = MR.f_inf().full_reduce()
>>> E2    = MR.E2().full_reduce()
>>> E4    = MR.E4().full_reduce()
>>> rho   = MR.group().rho()

>>> f_rho(rho)
0
>>> f_rho(rho + RealNumber('1e-100'))    # since rho == rho + 1e-100
0
>>> f_rho(rho + RealNumber('1e-6'))
2.525...e-10 - 3.884...e-6*I
>>> f_i(i)
0
>>> f_i(i + RealNumber('1e-1000'))  # rel tol 5e-2
-6.08402217494586e-14 - 4.10147008296517e-1000*I
>>> f_inf(infinity)
0

>>> i = I = QuadraticField(-Integer(1), 'I').gen()
>>> z = -Integer(1)/(-Integer(1)/(Integer(2)*i+Integer(30))-Integer(1))
>>> z
2/965*I + 934/965
>>> E4(z)
32288.05588811... - 118329.8566016...*I
>>> E4(z, prec=Integer(30), num_prec=Integer(100))    # long time
32288.0558872351130041311053... - 118329.856600349999751420381...*I
>>> E2(z)
409.3144737105... + 100.6926857489...*I
>>> E2(z, prec=Integer(30), num_prec=Integer(100))    # long time
409.314473710489761254584951... + 100.692685748952440684513866...*I
>>> (E2**Integer(2)-E4)(z)
125111.2655383... + 200759.8039479...*I
>>> (E2**Integer(2)-E4)(z, prec=Integer(30), num_prec=Integer(100))    # long time
125111.265538336196262200469... + 200759.803948009905410385699...*I

>>> (E2**Integer(2)-E4)(infinity)
0
>>> (Integer(1)/(E2**Integer(2)-E4))(infinity)
+Infinity
>>> ((E2**Integer(2)-E4)/f_inf)(infinity)
-3/(10*d)

>>> G = HeckeTriangleGroup(n=Integer(8))
>>> MR = QuasiMeromorphicModularFormsRing(group=G, red_hom=True)
>>> f_rho = MR.f_rho().full_reduce()
>>> f_i   = MR.f_i().full_reduce()
>>> E2    = MR.E2().full_reduce()

>>> z = AlgebraicField()(Integer(1)/Integer(10)+Integer(13)/Integer(10)*I)
>>> A = G.V(Integer(4))
>>> S = G.S()
>>> T = G.T()
>>> A == (T*S)**Integer(3)*T
True
>>> az = A.acton(z)
>>> az == (A[Integer(0),Integer(0)]*z + A[Integer(0),Integer(1)]) / (A[Integer(1),Integer(0)]*z + A[Integer(1),Integer(1)])
True

>>> f_rho(z)
1.03740476727... + 0.0131941034523...*I
>>> f_rho(az)
-2.29216470688... - 1.46235057536...*I
>>> k = f_rho.weight()
>>> aut_fact = f_rho.ep()**Integer(3) * (((T*S)**Integer(2)*T).acton(z)/AlgebraicField()(i))**k * (((T*S)*T).acton(z)/AlgebraicField()(i))**k * (T.acton(z)/AlgebraicField()(i))**k
>>> abs(aut_fact - f_rho.parent().aut_factor(A, z)) < RealNumber('1e-12')
True
>>> aut_fact * f_rho(z)
-2.29216470688... - 1.46235057536...*I

>>> f_rho.parent().default_num_prec(Integer(1000))
>>> f_rho.parent().default_prec(Integer(300))
>>> (f_rho.q_expansion_fixed_d().polynomial())(exp((Integer(2)*pi*i).n(Integer(1000))*z/G.lam()))    # long time
1.0374047672719462149821251... + 0.013194103452368974597290332...*I
>>> (f_rho.q_expansion_fixed_d().polynomial())(exp((Integer(2)*pi*i).n(Integer(1000))*az/G.lam()))    # long time
-2.2921647068881834598616367... - 1.4623505753697635207183406...*I

>>> f_i(z)
0.667489320423... - 0.118902824870...*I
>>> f_i(az)
14.5845388476... - 28.4604652892...*I
>>> k = f_i.weight()
>>> aut_fact = f_i.ep()**Integer(3) * (((T*S)**Integer(2)*T).acton(z)/AlgebraicField()(i))**k * (((T*S)*T).acton(z)/AlgebraicField()(i))**k * (T.acton(z)/AlgebraicField()(i))**k
>>> abs(aut_fact - f_i.parent().aut_factor(A, z)) < RealNumber('1e-12')
True
>>> aut_fact * f_i(z)
14.5845388476... - 28.4604652892...*I

>>> f_i.parent().default_num_prec(Integer(1000))
>>> f_i.parent().default_prec(Integer(300))
>>> (f_i.q_expansion_fixed_d().polynomial())(exp((Integer(2)*pi*i).n(Integer(1000))*z/G.lam()))    # long time
0.66748932042300250077433252... - 0.11890282487028677063054267...*I
>>> (f_i.q_expansion_fixed_d().polynomial())(exp((Integer(2)*pi*i).n(Integer(1000))*az/G.lam()))    # long time
14.584538847698600875918891... - 28.460465289220303834894855...*I

>>> f = f_rho*E2
>>> f(z)
0.966024386418... - 0.0138894699429...*I
>>> f(az)
-15.9978074989... - 29.2775758341...*I
>>> k = f.weight()
>>> aut_fact = f.ep()**Integer(3) * (((T*S)**Integer(2)*T).acton(z)/AlgebraicField()(i))**k * (((T*S)*T).acton(z)/AlgebraicField()(i))**k * (T.acton(z)/AlgebraicField()(i))**k
>>> abs(aut_fact - f.parent().aut_factor(A, z)) < RealNumber('1e-12')
True
>>> k2 = f_rho.weight()
>>> aut_fact2 = f_rho.ep() * (((T*S)**Integer(2)*T).acton(z)/AlgebraicField()(i))**k2 * (((T*S)*T).acton(z)/AlgebraicField()(i))**k2 * (T.acton(z)/AlgebraicField()(i))**k2
>>> abs(aut_fact2 - f_rho.parent().aut_factor(A, z)) < RealNumber('1e-12')
True
>>> cor_term = (Integer(4) * G.n() / (G.n()-Integer(2)) * A.c() * (A.c()*z+A.d())) / (Integer(2)*pi*i).n(Integer(1000)) * G.lam()
>>> aut_fact*f(z) + cor_term*aut_fact2*f_rho(z)
-15.9978074989... - 29.2775758341...*I

>>> f.parent().default_num_prec(Integer(1000))
>>> f.parent().default_prec(Integer(300))
>>> (f.q_expansion_fixed_d().polynomial())(exp((Integer(2)*pi*i).n(Integer(1000))*z/G.lam()))    # long time
0.96602438641867296777809436... - 0.013889469942995530807311503...*I
>>> (f.q_expansion_fixed_d().polynomial())(exp((Integer(2)*pi*i).n(Integer(1000))*az/G.lam()))    # long time
-15.997807498958825352887040... - 29.277575834123246063432206...*I

>>> MR = QuasiMeromorphicModularFormsRing(n=infinity, red_hom=True)
>>> f_i   = MR.f_i().full_reduce()
>>> f_inf = MR.f_inf().full_reduce()
>>> E2    = MR.E2().full_reduce()
>>> E4    = MR.E4().full_reduce()

>>> f_i(i)
0
>>> f_i(i + RealNumber('1e-1000'))
2.991...e-12 - 3.048...e-1000*I
>>> f_inf(infinity)
0

>>> z = -Integer(1)/(-Integer(1)/(Integer(2)*i+Integer(30))-Integer(1))
>>> E4(z, prec=Integer(15))
804.0722034... + 211.9278206...*I
>>> E4(z, prec=Integer(30), num_prec=Integer(100))    # long time
803.928382417... + 211.889914044...*I
>>> E2(z)
2.438455612... - 39.48442265...*I
>>> E2(z, prec=Integer(30), num_prec=Integer(100))    # long time
2.43968197227756036957475... - 39.4842637577742677851431...*I
>>> (E2**Integer(2)-E4)(z)
-2265.442515... - 380.3197877...*I
>>> (E2**Integer(2)-E4)(z, prec=Integer(30), num_prec=Integer(100))    # long time
-2265.44251550679807447320... - 380.319787790548788238792...*I

>>> (E2**Integer(2)-E4)(infinity)
0
>>> (Integer(1)/(E2**Integer(2)-E4))(infinity)
+Infinity
>>> ((E2**Integer(2)-E4)/f_inf)(infinity)
-1/(2*d)

>>> G = HeckeTriangleGroup(n=Infinity)
>>> z = AlgebraicField()(Integer(1)/Integer(10)+Integer(13)/Integer(10)*I)
>>> A = G.V(Integer(4))
>>> S = G.S()
>>> T = G.T()
>>> A == (T*S)**Integer(3)*T
True
>>> az = A.acton(z)
>>> az == (A[Integer(0),Integer(0)]*z + A[Integer(0),Integer(1)]) / (A[Integer(1),Integer(0)]*z + A[Integer(1),Integer(1)])
True

>>> f_i(z)
0.6208853409... - 0.1212525492...*I
>>> f_i(az)
6.103314419... + 20.42678597...*I
>>> k = f_i.weight()
>>> aut_fact = f_i.ep()**Integer(3) * (((T*S)**Integer(2)*T).acton(z)/AlgebraicField()(i))**k * (((T*S)*T).acton(z)/AlgebraicField()(i))**k * (T.acton(z)/AlgebraicField()(i))**k
>>> abs(aut_fact - f_i.parent().aut_factor(A, z)) < RealNumber('1e-12')
True
>>> aut_fact * f_i(z)
6.103314419... + 20.42678597...*I

>>> f_i.parent().default_num_prec(Integer(1000))
>>> f_i.parent().default_prec(Integer(300))
>>> (f_i.q_expansion_fixed_d().polynomial())(exp((Integer(2)*pi*i).n(Integer(1000))*z/G.lam()))    # long time
0.620885340917559158572271... - 0.121252549240996430425967...*I
>>> (f_i.q_expansion_fixed_d().polynomial())(exp((Integer(2)*pi*i).n(Integer(1000))*az/G.lam()))    # long time
6.10331441975198186745017... + 20.4267859728657976382684...*I

>>> f = f_i*E2
>>> f(z)
0.5349190275... - 0.1322370856...*I
>>> f(az)
-140.4711702... + 469.0793692...*I
>>> k = f.weight()
>>> aut_fact = f.ep()**Integer(3) * (((T*S)**Integer(2)*T).acton(z)/AlgebraicField()(i))**k * (((T*S)*T).acton(z)/AlgebraicField()(i))**k * (T.acton(z)/AlgebraicField()(i))**k
>>> abs(aut_fact - f.parent().aut_factor(A, z)) < RealNumber('1e-12')
True
>>> k2 = f_i.weight()
>>> aut_fact2 = f_i.ep() * (((T*S)**Integer(2)*T).acton(z)/AlgebraicField()(i))**k2 * (((T*S)*T).acton(z)/AlgebraicField()(i))**k2 * (T.acton(z)/AlgebraicField()(i))**k2
>>> abs(aut_fact2 - f_i.parent().aut_factor(A, z)) < RealNumber('1e-12')
True
>>> cor_term = (Integer(4) * A.c() * (A.c()*z+A.d())) / (Integer(2)*pi*i).n(Integer(1000)) * G.lam()
>>> aut_fact*f(z) + cor_term*aut_fact2*f_i(z)
-140.4711702... + 469.0793692...*I

>>> f.parent().default_num_prec(Integer(1000))
>>> f.parent().default_prec(Integer(300))
>>> (f.q_expansion_fixed_d().polynomial())(exp((Integer(2)*pi*i).n(Integer(1000))*z/G.lam()))    # long time
0.534919027587592616802582... - 0.132237085641931661668338...*I

>>> (f.q_expansion_fixed_d().polynomial())(exp((Integer(2)*pi*i).n(Integer(1000))*az/G.lam()))    # long time
-140.471170232432551196978... + 469.079369280804086032719...*I

Sage Live

from sage.modular.modform_hecketriangle.hecke_triangle_groups import HeckeTriangleGroup
from sage.modular.modform_hecketriangle.graded_ring import QuasiMeromorphicModularFormsRing
MR = QuasiMeromorphicModularFormsRing(n=5, red_hom=True)
f_rho = MR.f_rho().full_reduce()
f_i   = MR.f_i().full_reduce()
f_inf = MR.f_inf().full_reduce()
E2    = MR.E2().full_reduce()
E4    = MR.E4().full_reduce()
rho   = MR.group().rho()
f_rho(rho)
f_rho(rho + 1e-100)    # since rho == rho + 1e-100
f_rho(rho + 1e-6)
f_i(i)
f_i(i + 1e-1000)  # rel tol 5e-2
f_inf(infinity)
i = I = QuadraticField(-1, 'I').gen()
z = -1/(-1/(2*i+30)-1)
z
E4(z)
E4(z, prec=30, num_prec=100)    # long time
E2(z)
E2(z, prec=30, num_prec=100)    # long time
(E2^2-E4)(z)
(E2^2-E4)(z, prec=30, num_prec=100)    # long time
(E2^2-E4)(infinity)
(1/(E2^2-E4))(infinity)
((E2^2-E4)/f_inf)(infinity)
G = HeckeTriangleGroup(n=8)
MR = QuasiMeromorphicModularFormsRing(group=G, red_hom=True)
f_rho = MR.f_rho().full_reduce()
f_i   = MR.f_i().full_reduce()
E2    = MR.E2().full_reduce()
z = AlgebraicField()(1/10+13/10*I)
A = G.V(4)
S = G.S()
T = G.T()
A == (T*S)**3*T
az = A.acton(z)
az == (A[0,0]*z + A[0,1]) / (A[1,0]*z + A[1,1])
f_rho(z)
f_rho(az)
k = f_rho.weight()
aut_fact = f_rho.ep()^3 * (((T*S)**2*T).acton(z)/AlgebraicField()(i))**k * (((T*S)*T).acton(z)/AlgebraicField()(i))**k * (T.acton(z)/AlgebraicField()(i))**k
abs(aut_fact - f_rho.parent().aut_factor(A, z)) < 1e-12
aut_fact * f_rho(z)
f_rho.parent().default_num_prec(1000)
f_rho.parent().default_prec(300)
(f_rho.q_expansion_fixed_d().polynomial())(exp((2*pi*i).n(1000)*z/G.lam()))    # long time
(f_rho.q_expansion_fixed_d().polynomial())(exp((2*pi*i).n(1000)*az/G.lam()))    # long time
f_i(z)
f_i(az)
k = f_i.weight()
aut_fact = f_i.ep()^3 * (((T*S)**2*T).acton(z)/AlgebraicField()(i))**k * (((T*S)*T).acton(z)/AlgebraicField()(i))**k * (T.acton(z)/AlgebraicField()(i))**k
abs(aut_fact - f_i.parent().aut_factor(A, z)) < 1e-12
aut_fact * f_i(z)
f_i.parent().default_num_prec(1000)
f_i.parent().default_prec(300)
(f_i.q_expansion_fixed_d().polynomial())(exp((2*pi*i).n(1000)*z/G.lam()))    # long time
(f_i.q_expansion_fixed_d().polynomial())(exp((2*pi*i).n(1000)*az/G.lam()))    # long time
f = f_rho*E2
f(z)
f(az)
k = f.weight()
aut_fact = f.ep()^3 * (((T*S)**2*T).acton(z)/AlgebraicField()(i))**k * (((T*S)*T).acton(z)/AlgebraicField()(i))**k * (T.acton(z)/AlgebraicField()(i))**k
abs(aut_fact - f.parent().aut_factor(A, z)) < 1e-12
k2 = f_rho.weight()
aut_fact2 = f_rho.ep() * (((T*S)**2*T).acton(z)/AlgebraicField()(i))**k2 * (((T*S)*T).acton(z)/AlgebraicField()(i))**k2 * (T.acton(z)/AlgebraicField()(i))**k2
abs(aut_fact2 - f_rho.parent().aut_factor(A, z)) < 1e-12
cor_term = (4 * G.n() / (G.n()-2) * A.c() * (A.c()*z+A.d())) / (2*pi*i).n(1000) * G.lam()
aut_fact*f(z) + cor_term*aut_fact2*f_rho(z)
f.parent().default_num_prec(1000)
f.parent().default_prec(300)
(f.q_expansion_fixed_d().polynomial())(exp((2*pi*i).n(1000)*z/G.lam()))    # long time
(f.q_expansion_fixed_d().polynomial())(exp((2*pi*i).n(1000)*az/G.lam()))    # long time
MR = QuasiMeromorphicModularFormsRing(n=infinity, red_hom=True)
f_i   = MR.f_i().full_reduce()
f_inf = MR.f_inf().full_reduce()
E2    = MR.E2().full_reduce()
E4    = MR.E4().full_reduce()
f_i(i)
f_i(i + 1e-1000)
f_inf(infinity)
z = -1/(-1/(2*i+30)-1)
E4(z, prec=15)
E4(z, prec=30, num_prec=100)    # long time
E2(z)
E2(z, prec=30, num_prec=100)    # long time
(E2^2-E4)(z)
(E2^2-E4)(z, prec=30, num_prec=100)    # long time
(E2^2-E4)(infinity)
(1/(E2^2-E4))(infinity)
((E2^2-E4)/f_inf)(infinity)
G = HeckeTriangleGroup(n=Infinity)
z = AlgebraicField()(1/10+13/10*I)
A = G.V(4)
S = G.S()
T = G.T()
A == (T*S)**3*T
az = A.acton(z)
az == (A[0,0]*z + A[0,1]) / (A[1,0]*z + A[1,1])
f_i(z)
f_i(az)
k = f_i.weight()
aut_fact = f_i.ep()^3 * (((T*S)**2*T).acton(z)/AlgebraicField()(i))**k * (((T*S)*T).acton(z)/AlgebraicField()(i))**k * (T.acton(z)/AlgebraicField()(i))**k
abs(aut_fact - f_i.parent().aut_factor(A, z)) < 1e-12
aut_fact * f_i(z)
f_i.parent().default_num_prec(1000)
f_i.parent().default_prec(300)
(f_i.q_expansion_fixed_d().polynomial())(exp((2*pi*i).n(1000)*z/G.lam()))    # long time
(f_i.q_expansion_fixed_d().polynomial())(exp((2*pi*i).n(1000)*az/G.lam()))    # long time
f = f_i*E2
f(z)
f(az)
k = f.weight()
aut_fact = f.ep()^3 * (((T*S)**2*T).acton(z)/AlgebraicField()(i))**k * (((T*S)*T).acton(z)/AlgebraicField()(i))**k * (T.acton(z)/AlgebraicField()(i))**k
abs(aut_fact - f.parent().aut_factor(A, z)) < 1e-12
k2 = f_i.weight()
aut_fact2 = f_i.ep() * (((T*S)**2*T).acton(z)/AlgebraicField()(i))**k2 * (((T*S)*T).acton(z)/AlgebraicField()(i))**k2 * (T.acton(z)/AlgebraicField()(i))**k2
abs(aut_fact2 - f_i.parent().aut_factor(A, z)) < 1e-12
cor_term = (4 * A.c() * (A.c()*z+A.d())) / (2*pi*i).n(1000) * G.lam()
aut_fact*f(z) + cor_term*aut_fact2*f_i(z)
f.parent().default_num_prec(1000)
f.parent().default_prec(300)
(f.q_expansion_fixed_d().polynomial())(exp((2*pi*i).n(1000)*z/G.lam()))    # long time
(f.q_expansion_fixed_d().polynomial())(exp((2*pi*i).n(1000)*az/G.lam()))    # long time

It is possible to evaluate at points of HyperbolicPlane:

Sage

sage: # needs sage.symbolic
sage: p = HyperbolicPlane().PD().get_point(-I/2)
sage: bool(p.to_model('UHP').coordinates() == I/3)
True
sage: E4(p) == E4(I/3)
True
sage: p = HyperbolicPlane().PD().get_point(I)
sage: f_inf(p, check=True) == 0
True
sage: (1/(E2^2-E4))(p) == infinity
True

Python

>>> from sage.all import *
>>> # needs sage.symbolic
>>> p = HyperbolicPlane().PD().get_point(-I/Integer(2))
>>> bool(p.to_model('UHP').coordinates() == I/Integer(3))
True
>>> E4(p) == E4(I/Integer(3))
True
>>> p = HyperbolicPlane().PD().get_point(I)
>>> f_inf(p, check=True) == Integer(0)
True
>>> (Integer(1)/(E2**Integer(2)-E4))(p) == infinity
True

Sage Live

# needs sage.symbolic
p = HyperbolicPlane().PD().get_point(-I/2)
bool(p.to_model('UHP').coordinates() == I/3)
E4(p) == E4(I/3)
p = HyperbolicPlane().PD().get_point(I)
f_inf(p, check=True) == 0
(1/(E2^2-E4))(p) == infinity

full_reduce()[source]¶

Convert self into its reduced parent.

EXAMPLES:

Sage

sage: from sage.modular.modform_hecketriangle.graded_ring import QuasiMeromorphicModularFormsRing
sage: Delta = QuasiMeromorphicModularFormsRing().Delta()
sage: Delta
f_rho^3*d - f_i^2*d
sage: Delta.full_reduce()
q - 24*q^2 + 252*q^3 - 1472*q^4 + O(q^5)
sage: Delta.full_reduce().parent() == Delta.reduced_parent()
True

sage: QuasiMeromorphicModularFormsRing().Delta().full_reduce().parent()
CuspForms(n=3, k=12, ep=1) over Integer Ring

Python

>>> from sage.all import *
>>> from sage.modular.modform_hecketriangle.graded_ring import QuasiMeromorphicModularFormsRing
>>> Delta = QuasiMeromorphicModularFormsRing().Delta()
>>> Delta
f_rho^3*d - f_i^2*d
>>> Delta.full_reduce()
q - 24*q^2 + 252*q^3 - 1472*q^4 + O(q^5)
>>> Delta.full_reduce().parent() == Delta.reduced_parent()
True

>>> QuasiMeromorphicModularFormsRing().Delta().full_reduce().parent()
CuspForms(n=3, k=12, ep=1) over Integer Ring

Sage Live

from sage.modular.modform_hecketriangle.graded_ring import QuasiMeromorphicModularFormsRing
Delta = QuasiMeromorphicModularFormsRing().Delta()
Delta
Delta.full_reduce()
Delta.full_reduce().parent() == Delta.reduced_parent()
QuasiMeromorphicModularFormsRing().Delta().full_reduce().parent()

group()[source]¶

Return the (Hecke triangle) group of self.parent().

EXAMPLES:

Sage

sage: from sage.modular.modform_hecketriangle.space import ModularForms
sage: ModularForms(n=12, k=4).E4().group()
Hecke triangle group for n = 12

Python

>>> from sage.all import *
>>> from sage.modular.modform_hecketriangle.space import ModularForms
>>> ModularForms(n=Integer(12), k=Integer(4)).E4().group()
Hecke triangle group for n = 12

Sage Live

from sage.modular.modform_hecketriangle.space import ModularForms
ModularForms(n=12, k=4).E4().group()

hecke_n()[source]¶

Return the parameter n of the (Hecke triangle) group of self.parent().

EXAMPLES:

Sage

sage: from sage.modular.modform_hecketriangle.space import ModularForms
sage: ModularForms(n=12, k=6).E6().hecke_n()
12

Python

>>> from sage.all import *
>>> from sage.modular.modform_hecketriangle.space import ModularForms
>>> ModularForms(n=Integer(12), k=Integer(6)).E6().hecke_n()
12

Sage Live

from sage.modular.modform_hecketriangle.space import ModularForms
ModularForms(n=12, k=6).E6().hecke_n()

is_cuspidal()[source]¶

Return whether self is cuspidal in the sense that self is holomorphic and f_inf divides the numerator.

EXAMPLES:

Sage

sage: from sage.modular.modform_hecketriangle.graded_ring import QuasiModularFormsRing
sage: from sage.modular.modform_hecketriangle.space import QuasiModularForms
sage: x, y, z, d = var("x,y,z,d")                                           # needs sage.symbolic
sage: QuasiModularFormsRing(n=5)(y^3-z^5).is_cuspidal()                     # needs sage.symbolic
False
sage: QuasiModularFormsRing(n=5)(z*x^5-z*y^2).is_cuspidal()                 # needs sage.symbolic
True
sage: QuasiModularForms(n=18).Delta().is_cuspidal()
True
sage: QuasiModularForms(n=18).f_rho().is_cuspidal()
False
sage: QuasiModularForms(n=infinity).f_inf().is_cuspidal()
False
sage: QuasiModularForms(n=infinity).Delta().is_cuspidal()
True

Python

>>> from sage.all import *
>>> from sage.modular.modform_hecketriangle.graded_ring import QuasiModularFormsRing
>>> from sage.modular.modform_hecketriangle.space import QuasiModularForms
>>> x, y, z, d = var("x,y,z,d")                                           # needs sage.symbolic
>>> QuasiModularFormsRing(n=Integer(5))(y**Integer(3)-z**Integer(5)).is_cuspidal()                     # needs sage.symbolic
False
>>> QuasiModularFormsRing(n=Integer(5))(z*x**Integer(5)-z*y**Integer(2)).is_cuspidal()                 # needs sage.symbolic
True
>>> QuasiModularForms(n=Integer(18)).Delta().is_cuspidal()
True
>>> QuasiModularForms(n=Integer(18)).f_rho().is_cuspidal()
False
>>> QuasiModularForms(n=infinity).f_inf().is_cuspidal()
False
>>> QuasiModularForms(n=infinity).Delta().is_cuspidal()
True

Sage Live

from sage.modular.modform_hecketriangle.graded_ring import QuasiModularFormsRing
from sage.modular.modform_hecketriangle.space import QuasiModularForms
x, y, z, d = var("x,y,z,d")                                           # needs sage.symbolic
QuasiModularFormsRing(n=5)(y^3-z^5).is_cuspidal()                     # needs sage.symbolic
QuasiModularFormsRing(n=5)(z*x^5-z*y^2).is_cuspidal()                 # needs sage.symbolic
QuasiModularForms(n=18).Delta().is_cuspidal()
QuasiModularForms(n=18).f_rho().is_cuspidal()
QuasiModularForms(n=infinity).f_inf().is_cuspidal()
QuasiModularForms(n=infinity).Delta().is_cuspidal()

is_holomorphic()[source]¶

Return whether self is holomorphic in the sense that the denominator of self is constant.

EXAMPLES:

Sage

sage: from sage.modular.modform_hecketriangle.graded_ring import QuasiMeromorphicModularFormsRing
sage: from sage.modular.modform_hecketriangle.space import QuasiMeromorphicModularForms
sage: x, y, z, d = var("x,y,z,d")                                                       # needs sage.symbolic
sage: QuasiMeromorphicModularFormsRing(n=5)((y^3-z^5)/(x^5-y^2)+y-d).is_holomorphic()   # needs sage.symbolic
False
sage: QuasiMeromorphicModularFormsRing(n=5)(x^2+y-d+z).is_holomorphic()                 # needs sage.symbolic
True
sage: QuasiMeromorphicModularForms(n=18).J_inv().is_holomorphic()
False
sage: QuasiMeromorphicModularForms(n=18).f_i().is_holomorphic()
True
sage: QuasiMeromorphicModularForms(n=infinity).f_inf().is_holomorphic()
True

Python

>>> from sage.all import *
>>> from sage.modular.modform_hecketriangle.graded_ring import QuasiMeromorphicModularFormsRing
>>> from sage.modular.modform_hecketriangle.space import QuasiMeromorphicModularForms
>>> x, y, z, d = var("x,y,z,d")                                                       # needs sage.symbolic
>>> QuasiMeromorphicModularFormsRing(n=Integer(5))((y**Integer(3)-z**Integer(5))/(x**Integer(5)-y**Integer(2))+y-d).is_holomorphic()   # needs sage.symbolic
False
>>> QuasiMeromorphicModularFormsRing(n=Integer(5))(x**Integer(2)+y-d+z).is_holomorphic()                 # needs sage.symbolic
True
>>> QuasiMeromorphicModularForms(n=Integer(18)).J_inv().is_holomorphic()
False
>>> QuasiMeromorphicModularForms(n=Integer(18)).f_i().is_holomorphic()
True
>>> QuasiMeromorphicModularForms(n=infinity).f_inf().is_holomorphic()
True

Sage Live

from sage.modular.modform_hecketriangle.graded_ring import QuasiMeromorphicModularFormsRing
from sage.modular.modform_hecketriangle.space import QuasiMeromorphicModularForms
x, y, z, d = var("x,y,z,d")                                                       # needs sage.symbolic
QuasiMeromorphicModularFormsRing(n=5)((y^3-z^5)/(x^5-y^2)+y-d).is_holomorphic()   # needs sage.symbolic
QuasiMeromorphicModularFormsRing(n=5)(x^2+y-d+z).is_holomorphic()                 # needs sage.symbolic
QuasiMeromorphicModularForms(n=18).J_inv().is_holomorphic()
QuasiMeromorphicModularForms(n=18).f_i().is_holomorphic()
QuasiMeromorphicModularForms(n=infinity).f_inf().is_holomorphic()

is_homogeneous()[source]¶

Return whether self is homogeneous.

EXAMPLES:

Sage

sage: from sage.modular.modform_hecketriangle.graded_ring import QuasiModularFormsRing
sage: QuasiModularFormsRing(n=12).Delta().is_homogeneous()
True
sage: QuasiModularFormsRing(n=12).Delta().parent().is_homogeneous()
False
sage: x, y, z, d = var("x,y,z,d")                                           # needs sage.symbolic
sage: QuasiModularFormsRing(n=12)(x^3+y^2+z+d).is_homogeneous()             # needs sage.symbolic
False

sage: QuasiModularFormsRing(n=infinity)(x*(x-y^2)+y^4).is_homogeneous()     # needs sage.symbolic
True

Python

>>> from sage.all import *
>>> from sage.modular.modform_hecketriangle.graded_ring import QuasiModularFormsRing
>>> QuasiModularFormsRing(n=Integer(12)).Delta().is_homogeneous()
True
>>> QuasiModularFormsRing(n=Integer(12)).Delta().parent().is_homogeneous()
False
>>> x, y, z, d = var("x,y,z,d")                                           # needs sage.symbolic
>>> QuasiModularFormsRing(n=Integer(12))(x**Integer(3)+y**Integer(2)+z+d).is_homogeneous()             # needs sage.symbolic
False

>>> QuasiModularFormsRing(n=infinity)(x*(x-y**Integer(2))+y**Integer(4)).is_homogeneous()     # needs sage.symbolic
True

Sage Live

from sage.modular.modform_hecketriangle.graded_ring import QuasiModularFormsRing
QuasiModularFormsRing(n=12).Delta().is_homogeneous()
QuasiModularFormsRing(n=12).Delta().parent().is_homogeneous()
x, y, z, d = var("x,y,z,d")                                           # needs sage.symbolic
QuasiModularFormsRing(n=12)(x^3+y^2+z+d).is_homogeneous()             # needs sage.symbolic
QuasiModularFormsRing(n=infinity)(x*(x-y^2)+y^4).is_homogeneous()     # needs sage.symbolic

is_modular()[source]¶

Return whether self (resp. its homogeneous components) transform like modular forms.

EXAMPLES:

Sage

sage: from sage.modular.modform_hecketriangle.graded_ring import QuasiModularFormsRing
sage: from sage.modular.modform_hecketriangle.space import QuasiModularForms
sage: x, y, z, d = var("x,y,z,d")                                           # needs sage.symbolic
sage: QuasiModularFormsRing(n=5)(x^2+y-d).is_modular()                      # needs sage.symbolic
True
sage: QuasiModularFormsRing(n=5)(x^2+y-d+z).is_modular()                    # needs sage.symbolic
False
sage: QuasiModularForms(n=18).f_i().is_modular()
True
sage: QuasiModularForms(n=18).E2().is_modular()
False
sage: QuasiModularForms(n=infinity).f_inf().is_modular()
True

Python

>>> from sage.all import *
>>> from sage.modular.modform_hecketriangle.graded_ring import QuasiModularFormsRing
>>> from sage.modular.modform_hecketriangle.space import QuasiModularForms
>>> x, y, z, d = var("x,y,z,d")                                           # needs sage.symbolic
>>> QuasiModularFormsRing(n=Integer(5))(x**Integer(2)+y-d).is_modular()                      # needs sage.symbolic
True
>>> QuasiModularFormsRing(n=Integer(5))(x**Integer(2)+y-d+z).is_modular()                    # needs sage.symbolic
False
>>> QuasiModularForms(n=Integer(18)).f_i().is_modular()
True
>>> QuasiModularForms(n=Integer(18)).E2().is_modular()
False
>>> QuasiModularForms(n=infinity).f_inf().is_modular()
True

Sage Live

from sage.modular.modform_hecketriangle.graded_ring import QuasiModularFormsRing
from sage.modular.modform_hecketriangle.space import QuasiModularForms
x, y, z, d = var("x,y,z,d")                                           # needs sage.symbolic
QuasiModularFormsRing(n=5)(x^2+y-d).is_modular()                      # needs sage.symbolic
QuasiModularFormsRing(n=5)(x^2+y-d+z).is_modular()                    # needs sage.symbolic
QuasiModularForms(n=18).f_i().is_modular()
QuasiModularForms(n=18).E2().is_modular()
QuasiModularForms(n=infinity).f_inf().is_modular()

is_weakly_holomorphic()[source]¶

Return whether self is weakly holomorphic in the sense that: self has at most a power of f_inf in its denominator.

EXAMPLES:

Sage

sage: # needs sage.symbolic
sage: from sage.modular.modform_hecketriangle.graded_ring import QuasiMeromorphicModularFormsRing
sage: from sage.modular.modform_hecketriangle.space import QuasiMeromorphicModularForms
sage: x, y, z, d = var("x,y,z,d")
sage: QuasiMeromorphicModularFormsRing(n=5)(x/(x^5-y^2)+z).is_weakly_holomorphic()
True
sage: QuasiMeromorphicModularFormsRing(n=5)(x^2+y/x-d).is_weakly_holomorphic()
False
sage: QuasiMeromorphicModularForms(n=18).J_inv().is_weakly_holomorphic()
True
sage: QuasiMeromorphicModularForms(n=infinity, k=-4)(1/x).is_weakly_holomorphic()
True
sage: QuasiMeromorphicModularForms(n=infinity, k=-2)(1/y).is_weakly_holomorphic()
False

Python

>>> from sage.all import *
>>> # needs sage.symbolic
>>> from sage.modular.modform_hecketriangle.graded_ring import QuasiMeromorphicModularFormsRing
>>> from sage.modular.modform_hecketriangle.space import QuasiMeromorphicModularForms
>>> x, y, z, d = var("x,y,z,d")
>>> QuasiMeromorphicModularFormsRing(n=Integer(5))(x/(x**Integer(5)-y**Integer(2))+z).is_weakly_holomorphic()
True
>>> QuasiMeromorphicModularFormsRing(n=Integer(5))(x**Integer(2)+y/x-d).is_weakly_holomorphic()
False
>>> QuasiMeromorphicModularForms(n=Integer(18)).J_inv().is_weakly_holomorphic()
True
>>> QuasiMeromorphicModularForms(n=infinity, k=-Integer(4))(Integer(1)/x).is_weakly_holomorphic()
True
>>> QuasiMeromorphicModularForms(n=infinity, k=-Integer(2))(Integer(1)/y).is_weakly_holomorphic()
False

Sage Live

# needs sage.symbolic
from sage.modular.modform_hecketriangle.graded_ring import QuasiMeromorphicModularFormsRing
from sage.modular.modform_hecketriangle.space import QuasiMeromorphicModularForms
x, y, z, d = var("x,y,z,d")
QuasiMeromorphicModularFormsRing(n=5)(x/(x^5-y^2)+z).is_weakly_holomorphic()
QuasiMeromorphicModularFormsRing(n=5)(x^2+y/x-d).is_weakly_holomorphic()
QuasiMeromorphicModularForms(n=18).J_inv().is_weakly_holomorphic()
QuasiMeromorphicModularForms(n=infinity, k=-4)(1/x).is_weakly_holomorphic()
QuasiMeromorphicModularForms(n=infinity, k=-2)(1/y).is_weakly_holomorphic()

is_zero()[source]¶

Return whether self is the zero function.

EXAMPLES:

Sage

sage: from sage.modular.modform_hecketriangle.graded_ring import QuasiModularFormsRing
sage: from sage.modular.modform_hecketriangle.space import QuasiModularForms
sage: QuasiModularFormsRing(n=5)(1).is_zero()
False
sage: QuasiModularFormsRing(n=5)(0).is_zero()
True
sage: QuasiModularForms(n=18).zero().is_zero()
True
sage: QuasiModularForms(n=18).Delta().is_zero()
False
sage: QuasiModularForms(n=infinity).f_rho().is_zero()
False

Python

>>> from sage.all import *
>>> from sage.modular.modform_hecketriangle.graded_ring import QuasiModularFormsRing
>>> from sage.modular.modform_hecketriangle.space import QuasiModularForms
>>> QuasiModularFormsRing(n=Integer(5))(Integer(1)).is_zero()
False
>>> QuasiModularFormsRing(n=Integer(5))(Integer(0)).is_zero()
True
>>> QuasiModularForms(n=Integer(18)).zero().is_zero()
True
>>> QuasiModularForms(n=Integer(18)).Delta().is_zero()
False
>>> QuasiModularForms(n=infinity).f_rho().is_zero()
False

Sage Live

from sage.modular.modform_hecketriangle.graded_ring import QuasiModularFormsRing
from sage.modular.modform_hecketriangle.space import QuasiModularForms
QuasiModularFormsRing(n=5)(1).is_zero()
QuasiModularFormsRing(n=5)(0).is_zero()
QuasiModularForms(n=18).zero().is_zero()
QuasiModularForms(n=18).Delta().is_zero()
QuasiModularForms(n=infinity).f_rho().is_zero()

numerator()[source]¶

Return the numerator of self.

I.e. the (properly reduced) new form corresponding to the numerator of self.rat().

Note that the parent of self might (probably will) change.

EXAMPLES:

Sage

sage: # needs sage.symbolic
sage: from sage.modular.modform_hecketriangle.graded_ring import QuasiMeromorphicModularFormsRing
sage: from sage.modular.modform_hecketriangle.space import QuasiMeromorphicModularForms
sage: x, y, z, d = var("x,y,z,d")
sage: QuasiMeromorphicModularFormsRing(n=5)((y^3-z^5)/(x^5-y^2)+y-d).numerator()
f_rho^5*f_i - f_rho^5*d - E2^5 + f_i^2*d
sage: QuasiMeromorphicModularFormsRing(n=5)((y^3-z^5)/(x^5-y^2)+y-d).numerator().parent()
QuasiModularFormsRing(n=5) over Integer Ring
sage: QuasiMeromorphicModularForms(n=5, k=-2, ep=-1)(x/y).numerator()
1 + 7/(100*d)*q + 21/(160000*d^2)*q^2 + 1043/(192000000*d^3)*q^3 + 45479/(1228800000000*d^4)*q^4 + O(q^5)
sage: QuasiMeromorphicModularForms(n=5, k=-2, ep=-1)(x/y).numerator().parent()
QuasiModularForms(n=5, k=4/3, ep=1) over Integer Ring
sage: (QuasiMeromorphicModularForms(n=infinity, k=-2, ep=-1)(y/x)).numerator()
1 - 24*q + 24*q^2 - 96*q^3 + 24*q^4 + O(q^5)
sage: (QuasiMeromorphicModularForms(n=infinity, k=-2, ep=-1)(y/x)).numerator().parent()
QuasiModularForms(n=+Infinity, k=2, ep=-1) over Integer Ring

Python

>>> from sage.all import *
>>> # needs sage.symbolic
>>> from sage.modular.modform_hecketriangle.graded_ring import QuasiMeromorphicModularFormsRing
>>> from sage.modular.modform_hecketriangle.space import QuasiMeromorphicModularForms
>>> x, y, z, d = var("x,y,z,d")
>>> QuasiMeromorphicModularFormsRing(n=Integer(5))((y**Integer(3)-z**Integer(5))/(x**Integer(5)-y**Integer(2))+y-d).numerator()
f_rho^5*f_i - f_rho^5*d - E2^5 + f_i^2*d
>>> QuasiMeromorphicModularFormsRing(n=Integer(5))((y**Integer(3)-z**Integer(5))/(x**Integer(5)-y**Integer(2))+y-d).numerator().parent()
QuasiModularFormsRing(n=5) over Integer Ring
>>> QuasiMeromorphicModularForms(n=Integer(5), k=-Integer(2), ep=-Integer(1))(x/y).numerator()
1 + 7/(100*d)*q + 21/(160000*d^2)*q^2 + 1043/(192000000*d^3)*q^3 + 45479/(1228800000000*d^4)*q^4 + O(q^5)
>>> QuasiMeromorphicModularForms(n=Integer(5), k=-Integer(2), ep=-Integer(1))(x/y).numerator().parent()
QuasiModularForms(n=5, k=4/3, ep=1) over Integer Ring
>>> (QuasiMeromorphicModularForms(n=infinity, k=-Integer(2), ep=-Integer(1))(y/x)).numerator()
1 - 24*q + 24*q^2 - 96*q^3 + 24*q^4 + O(q^5)
>>> (QuasiMeromorphicModularForms(n=infinity, k=-Integer(2), ep=-Integer(1))(y/x)).numerator().parent()
QuasiModularForms(n=+Infinity, k=2, ep=-1) over Integer Ring

Sage Live

# needs sage.symbolic
from sage.modular.modform_hecketriangle.graded_ring import QuasiMeromorphicModularFormsRing
from sage.modular.modform_hecketriangle.space import QuasiMeromorphicModularForms
x, y, z, d = var("x,y,z,d")
QuasiMeromorphicModularFormsRing(n=5)((y^3-z^5)/(x^5-y^2)+y-d).numerator()
QuasiMeromorphicModularFormsRing(n=5)((y^3-z^5)/(x^5-y^2)+y-d).numerator().parent()
QuasiMeromorphicModularForms(n=5, k=-2, ep=-1)(x/y).numerator()
QuasiMeromorphicModularForms(n=5, k=-2, ep=-1)(x/y).numerator().parent()
(QuasiMeromorphicModularForms(n=infinity, k=-2, ep=-1)(y/x)).numerator()
(QuasiMeromorphicModularForms(n=infinity, k=-2, ep=-1)(y/x)).numerator().parent()

order_at(tau=+Infinity)[source]¶

Return the (overall) order of self at tau if easily possible: Namely if tau is infinity or congruent to i resp. rho.

It is possible to determine the order of points from HyperbolicPlane. In this case the coordinates of the upper half plane model are used.

If self is homogeneous and modular then the rational function self.rat() is used. Otherwise only tau=infinity is supported by using the Fourier expansion with increasing precision (until the order can be determined).

The function is mainly used to be able to work with the correct precision for Laurent series.

Note

For quasi forms one cannot deduce the analytic type from this order at infinity since the analytic order is defined by the behavior on each quasi part and not by their linear combination.

EXAMPLES:

Sage

sage: from sage.modular.modform_hecketriangle.graded_ring import QuasiMeromorphicModularFormsRing
sage: MR = QuasiMeromorphicModularFormsRing(red_hom=True)
sage: (MR.Delta()^3).order_at(infinity)
3
sage: MR.E2().order_at(infinity)
0
sage: (MR.J_inv()^2).order_at(infinity)
-2
sage: x,y,z,d = MR.pol_ring().gens()
sage: el = MR((z^3-y)^2/(x^3-y^2)).full_reduce()
sage: el
108*q + 11664*q^2 + 502848*q^3 + 12010464*q^4 + O(q^5)
sage: el.order_at(infinity)
1
sage: el.parent()
QuasiWeakModularForms(n=3, k=0, ep=1) over Integer Ring
sage: el.is_holomorphic()
False
sage: MR((z-y)^2+(x-y)^3).order_at(infinity)
2
sage: MR((x-y)^10).order_at(infinity)
10
sage: MR.zero().order_at(infinity)
+Infinity
sage: (MR(x*y^2)/MR.J_inv()).order_at(i)
2
sage: (MR(x*y^2)/MR.J_inv()).order_at(MR.group().rho())
-2

sage: MR = QuasiMeromorphicModularFormsRing(n=infinity, red_hom=True)
sage: (MR.Delta()^3*MR.E4()).order_at(infinity)
3
sage: MR.E2().order_at(infinity)
0
sage: (MR.J_inv()^2/MR.E4()).order_at(infinity)
-2
sage: el = MR((z^3-x*y)^2/(x^2*(x-y^2))).full_reduce()
sage: el
4*q - 304*q^2 + 8128*q^3 - 106144*q^4 + O(q^5)
sage: el.order_at(infinity)
1
sage: el.parent()
QuasiWeakModularForms(n=+Infinity, k=0, ep=1) over Integer Ring
sage: el.is_holomorphic()
False
sage: MR((z-x)^2+(x-y)^3).order_at(infinity)
2
sage: MR((x-y)^10).order_at(infinity)
10
sage: MR.zero().order_at(infinity)
+Infinity

sage: (MR.j_inv()*MR.f_i()^3).order_at(-1)
1
sage: (MR.j_inv()*MR.f_i()^3).order_at(i)
3
sage: (1/MR.f_inf()^2).order_at(-1)
0

sage: p = HyperbolicPlane().PD().get_point(I)                               # needs sage.symbolic
sage: MR((x-y)^10).order_at(p)                                              # needs sage.symbolic
10
sage: MR.zero().order_at(p)                                                 # needs sage.symbolic
+Infinity

Python

>>> from sage.all import *
>>> from sage.modular.modform_hecketriangle.graded_ring import QuasiMeromorphicModularFormsRing
>>> MR = QuasiMeromorphicModularFormsRing(red_hom=True)
>>> (MR.Delta()**Integer(3)).order_at(infinity)
3
>>> MR.E2().order_at(infinity)
0
>>> (MR.J_inv()**Integer(2)).order_at(infinity)
-2
>>> x,y,z,d = MR.pol_ring().gens()
>>> el = MR((z**Integer(3)-y)**Integer(2)/(x**Integer(3)-y**Integer(2))).full_reduce()
>>> el
108*q + 11664*q^2 + 502848*q^3 + 12010464*q^4 + O(q^5)
>>> el.order_at(infinity)
1
>>> el.parent()
QuasiWeakModularForms(n=3, k=0, ep=1) over Integer Ring
>>> el.is_holomorphic()
False
>>> MR((z-y)**Integer(2)+(x-y)**Integer(3)).order_at(infinity)
2
>>> MR((x-y)**Integer(10)).order_at(infinity)
10
>>> MR.zero().order_at(infinity)
+Infinity
>>> (MR(x*y**Integer(2))/MR.J_inv()).order_at(i)
2
>>> (MR(x*y**Integer(2))/MR.J_inv()).order_at(MR.group().rho())
-2

>>> MR = QuasiMeromorphicModularFormsRing(n=infinity, red_hom=True)
>>> (MR.Delta()**Integer(3)*MR.E4()).order_at(infinity)
3
>>> MR.E2().order_at(infinity)
0
>>> (MR.J_inv()**Integer(2)/MR.E4()).order_at(infinity)
-2
>>> el = MR((z**Integer(3)-x*y)**Integer(2)/(x**Integer(2)*(x-y**Integer(2)))).full_reduce()
>>> el
4*q - 304*q^2 + 8128*q^3 - 106144*q^4 + O(q^5)
>>> el.order_at(infinity)
1
>>> el.parent()
QuasiWeakModularForms(n=+Infinity, k=0, ep=1) over Integer Ring
>>> el.is_holomorphic()
False
>>> MR((z-x)**Integer(2)+(x-y)**Integer(3)).order_at(infinity)
2
>>> MR((x-y)**Integer(10)).order_at(infinity)
10
>>> MR.zero().order_at(infinity)
+Infinity

>>> (MR.j_inv()*MR.f_i()**Integer(3)).order_at(-Integer(1))
1
>>> (MR.j_inv()*MR.f_i()**Integer(3)).order_at(i)
3
>>> (Integer(1)/MR.f_inf()**Integer(2)).order_at(-Integer(1))
0

>>> p = HyperbolicPlane().PD().get_point(I)                               # needs sage.symbolic
>>> MR((x-y)**Integer(10)).order_at(p)                                              # needs sage.symbolic
10
>>> MR.zero().order_at(p)                                                 # needs sage.symbolic
+Infinity

Sage Live

from sage.modular.modform_hecketriangle.graded_ring import QuasiMeromorphicModularFormsRing
MR = QuasiMeromorphicModularFormsRing(red_hom=True)
(MR.Delta()^3).order_at(infinity)
MR.E2().order_at(infinity)
(MR.J_inv()^2).order_at(infinity)
x,y,z,d = MR.pol_ring().gens()
el = MR((z^3-y)^2/(x^3-y^2)).full_reduce()
el
el.order_at(infinity)
el.parent()
el.is_holomorphic()
MR((z-y)^2+(x-y)^3).order_at(infinity)
MR((x-y)^10).order_at(infinity)
MR.zero().order_at(infinity)
(MR(x*y^2)/MR.J_inv()).order_at(i)
(MR(x*y^2)/MR.J_inv()).order_at(MR.group().rho())
MR = QuasiMeromorphicModularFormsRing(n=infinity, red_hom=True)
(MR.Delta()^3*MR.E4()).order_at(infinity)
MR.E2().order_at(infinity)
(MR.J_inv()^2/MR.E4()).order_at(infinity)
el = MR((z^3-x*y)^2/(x^2*(x-y^2))).full_reduce()
el
el.order_at(infinity)
el.parent()
el.is_holomorphic()
MR((z-x)^2+(x-y)^3).order_at(infinity)
MR((x-y)^10).order_at(infinity)
MR.zero().order_at(infinity)
(MR.j_inv()*MR.f_i()^3).order_at(-1)
(MR.j_inv()*MR.f_i()^3).order_at(i)
(1/MR.f_inf()^2).order_at(-1)
p = HyperbolicPlane().PD().get_point(I)                               # needs sage.symbolic
MR((x-y)^10).order_at(p)                                              # needs sage.symbolic
MR.zero().order_at(p)                                                 # needs sage.symbolic

q_expansion(prec=None, fix_d=False, d_num_prec=None, fix_prec=False)[source]¶

Return the Fourier expansion of self.

INPUT:

prec – integer, the desired output precision O(q^prec). Default: None in which case the default precision of self.parent() is used.
fix_d – if False (default) a formal parameter is used for d. If True then the numerical value of d is used (resp. an exact value if the group is arithmetic). Otherwise the given value is used for d.
d_num_prec – the precision to be used if a numerical value for d is substituted (default: None), otherwise the default numerical precision of self.parent() is used
fix_prec – if fix_prec is not False (default) then the precision of the MFSeriesConstructor is increased such that the output has exactly the specified precision O(q^prec).

OUTPUT: the Fourier expansion of self as a FormalPowerSeries or FormalLaurentSeries

EXAMPLES:

Sage

sage: from sage.modular.modform_hecketriangle.graded_ring import WeakModularFormsRing, QuasiModularFormsRing
sage: j_inv = WeakModularFormsRing(red_hom=True).j_inv()
sage: j_inv.q_expansion(prec=3)
q^-1 + 31/(72*d) + 1823/(27648*d^2)*q + 10495/(2519424*d^3)*q^2 + O(q^3)

sage: E2 = QuasiModularFormsRing(n=5, red_hom=True).E2()
sage: E2.q_expansion(prec=3)
1 - 9/(200*d)*q - 369/(320000*d^2)*q^2 + O(q^3)
sage: E2.q_expansion(prec=3, fix_d=1)
1 - 9/200*q - 369/320000*q^2 + O(q^3)

sage: E6 = WeakModularFormsRing(n=5, red_hom=True).E6().full_reduce()
sage: Delta = WeakModularFormsRing(n=5, red_hom=True).Delta().full_reduce()
sage: E6.q_expansion(prec=3).prec() == 3
True
sage: (Delta/(E2^3-E6)).q_expansion(prec=3).prec() == 3
True
sage: (Delta/(E2^3-E6)^3).q_expansion(prec=3).prec() == 3
True
sage: ((E2^3-E6)/Delta^2).q_expansion(prec=3).prec() == 3
True
sage: ((E2^3-E6)^3/Delta).q_expansion(prec=3).prec() == 3
True

sage: x,y = var("x,y")
sage: el = WeakModularFormsRing()((x+1)/(x^3-y^2))
sage: el.q_expansion(prec=2, fix_prec = True)
2*d*q^-1 + O(1)
sage: el.q_expansion(prec=2)
2*d*q^-1 + 1/6 + 119/(41472*d)*q + O(q^2)

sage: j_inv = WeakModularFormsRing(n=infinity, red_hom=True).j_inv()
sage: j_inv.q_expansion(prec=3)
q^-1 + 3/(8*d) + 69/(1024*d^2)*q + 1/(128*d^3)*q^2 + O(q^3)

sage: E2 = QuasiModularFormsRing(n=infinity, red_hom=True).E2()
sage: E2.q_expansion(prec=3)
1 - 1/(8*d)*q - 1/(512*d^2)*q^2 + O(q^3)
sage: E2.q_expansion(prec=3, fix_d=1)
1 - 1/8*q - 1/512*q^2 + O(q^3)

sage: E4 = WeakModularFormsRing(n=infinity, red_hom=True).E4().full_reduce()
sage: Delta = WeakModularFormsRing(n=infinity, red_hom=True).Delta().full_reduce()
sage: E4.q_expansion(prec=3).prec() == 3
True
sage: (Delta/(E2^2-E4)).q_expansion(prec=3).prec() == 3
True
sage: (Delta/(E2^2-E4)^3).q_expansion(prec=3).prec() == 3
True
sage: ((E2^2-E4)/Delta^2).q_expansion(prec=3).prec() == 3
True
sage: ((E2^2-E4)^3/Delta).q_expansion(prec=3).prec() == 3
True

sage: x,y = var("x,y")
sage: el = WeakModularFormsRing(n=infinity)((x+1)/(x-y^2))
sage: el.q_expansion(prec=2, fix_prec = True)
2*d*q^-1 + O(1)
sage: el.q_expansion(prec=2)
2*d*q^-1 + 1/2 + 39/(512*d)*q + O(q^2)

Python

>>> from sage.all import *
>>> from sage.modular.modform_hecketriangle.graded_ring import WeakModularFormsRing, QuasiModularFormsRing
>>> j_inv = WeakModularFormsRing(red_hom=True).j_inv()
>>> j_inv.q_expansion(prec=Integer(3))
q^-1 + 31/(72*d) + 1823/(27648*d^2)*q + 10495/(2519424*d^3)*q^2 + O(q^3)

>>> E2 = QuasiModularFormsRing(n=Integer(5), red_hom=True).E2()
>>> E2.q_expansion(prec=Integer(3))
1 - 9/(200*d)*q - 369/(320000*d^2)*q^2 + O(q^3)
>>> E2.q_expansion(prec=Integer(3), fix_d=Integer(1))
1 - 9/200*q - 369/320000*q^2 + O(q^3)

>>> E6 = WeakModularFormsRing(n=Integer(5), red_hom=True).E6().full_reduce()
>>> Delta = WeakModularFormsRing(n=Integer(5), red_hom=True).Delta().full_reduce()
>>> E6.q_expansion(prec=Integer(3)).prec() == Integer(3)
True
>>> (Delta/(E2**Integer(3)-E6)).q_expansion(prec=Integer(3)).prec() == Integer(3)
True
>>> (Delta/(E2**Integer(3)-E6)**Integer(3)).q_expansion(prec=Integer(3)).prec() == Integer(3)
True
>>> ((E2**Integer(3)-E6)/Delta**Integer(2)).q_expansion(prec=Integer(3)).prec() == Integer(3)
True
>>> ((E2**Integer(3)-E6)**Integer(3)/Delta).q_expansion(prec=Integer(3)).prec() == Integer(3)
True

>>> x,y = var("x,y")
>>> el = WeakModularFormsRing()((x+Integer(1))/(x**Integer(3)-y**Integer(2)))
>>> el.q_expansion(prec=Integer(2), fix_prec = True)
2*d*q^-1 + O(1)
>>> el.q_expansion(prec=Integer(2))
2*d*q^-1 + 1/6 + 119/(41472*d)*q + O(q^2)

>>> j_inv = WeakModularFormsRing(n=infinity, red_hom=True).j_inv()
>>> j_inv.q_expansion(prec=Integer(3))
q^-1 + 3/(8*d) + 69/(1024*d^2)*q + 1/(128*d^3)*q^2 + O(q^3)

>>> E2 = QuasiModularFormsRing(n=infinity, red_hom=True).E2()
>>> E2.q_expansion(prec=Integer(3))
1 - 1/(8*d)*q - 1/(512*d^2)*q^2 + O(q^3)
>>> E2.q_expansion(prec=Integer(3), fix_d=Integer(1))
1 - 1/8*q - 1/512*q^2 + O(q^3)

>>> E4 = WeakModularFormsRing(n=infinity, red_hom=True).E4().full_reduce()
>>> Delta = WeakModularFormsRing(n=infinity, red_hom=True).Delta().full_reduce()
>>> E4.q_expansion(prec=Integer(3)).prec() == Integer(3)
True
>>> (Delta/(E2**Integer(2)-E4)).q_expansion(prec=Integer(3)).prec() == Integer(3)
True
>>> (Delta/(E2**Integer(2)-E4)**Integer(3)).q_expansion(prec=Integer(3)).prec() == Integer(3)
True
>>> ((E2**Integer(2)-E4)/Delta**Integer(2)).q_expansion(prec=Integer(3)).prec() == Integer(3)
True
>>> ((E2**Integer(2)-E4)**Integer(3)/Delta).q_expansion(prec=Integer(3)).prec() == Integer(3)
True

>>> x,y = var("x,y")
>>> el = WeakModularFormsRing(n=infinity)((x+Integer(1))/(x-y**Integer(2)))
>>> el.q_expansion(prec=Integer(2), fix_prec = True)
2*d*q^-1 + O(1)
>>> el.q_expansion(prec=Integer(2))
2*d*q^-1 + 1/2 + 39/(512*d)*q + O(q^2)

Sage Live

from sage.modular.modform_hecketriangle.graded_ring import WeakModularFormsRing, QuasiModularFormsRing
j_inv = WeakModularFormsRing(red_hom=True).j_inv()
j_inv.q_expansion(prec=3)
E2 = QuasiModularFormsRing(n=5, red_hom=True).E2()
E2.q_expansion(prec=3)
E2.q_expansion(prec=3, fix_d=1)
E6 = WeakModularFormsRing(n=5, red_hom=True).E6().full_reduce()
Delta = WeakModularFormsRing(n=5, red_hom=True).Delta().full_reduce()
E6.q_expansion(prec=3).prec() == 3
(Delta/(E2^3-E6)).q_expansion(prec=3).prec() == 3
(Delta/(E2^3-E6)^3).q_expansion(prec=3).prec() == 3
((E2^3-E6)/Delta^2).q_expansion(prec=3).prec() == 3
((E2^3-E6)^3/Delta).q_expansion(prec=3).prec() == 3
x,y = var("x,y")
el = WeakModularFormsRing()((x+1)/(x^3-y^2))
el.q_expansion(prec=2, fix_prec = True)
el.q_expansion(prec=2)
j_inv = WeakModularFormsRing(n=infinity, red_hom=True).j_inv()
j_inv.q_expansion(prec=3)
E2 = QuasiModularFormsRing(n=infinity, red_hom=True).E2()
E2.q_expansion(prec=3)
E2.q_expansion(prec=3, fix_d=1)
E4 = WeakModularFormsRing(n=infinity, red_hom=True).E4().full_reduce()
Delta = WeakModularFormsRing(n=infinity, red_hom=True).Delta().full_reduce()
E4.q_expansion(prec=3).prec() == 3
(Delta/(E2^2-E4)).q_expansion(prec=3).prec() == 3
(Delta/(E2^2-E4)^3).q_expansion(prec=3).prec() == 3
((E2^2-E4)/Delta^2).q_expansion(prec=3).prec() == 3
((E2^2-E4)^3/Delta).q_expansion(prec=3).prec() == 3
x,y = var("x,y")
el = WeakModularFormsRing(n=infinity)((x+1)/(x-y^2))
el.q_expansion(prec=2, fix_prec = True)
el.q_expansion(prec=2)

q_expansion_fixed_d(prec=None, d_num_prec=None, fix_prec=False)[source]¶

Return the Fourier expansion of self.

The numerical (or exact) value for d is substituted.

INPUT:

prec – integer; the desired output precision O(q^prec). Default: None, in which case the default precision of self.parent() is used.
d_num_prec – the precision to be used if a numerical value for d is substituted (default: None), otherwise the default numerical precision of self.parent() is used
fix_prec – if fix_prec is not False (default) then the precision of the MFSeriesConstructor is increased such that the output has exactly the specified precision O(q^prec).

OUTPUT: the Fourier expansion of self as a FormalPowerSeries or FormalLaurentSeries

EXAMPLES:

Sage

sage: from sage.modular.modform_hecketriangle.graded_ring import WeakModularFormsRing, QuasiModularFormsRing
sage: j_inv = WeakModularFormsRing(red_hom=True).j_inv()
sage: j_inv.q_expansion_fixed_d(prec=3)
q^-1 + 744 + 196884*q + 21493760*q^2 + O(q^3)

sage: E2 = QuasiModularFormsRing(n=5, red_hom=True).E2()
sage: E2.q_expansion_fixed_d(prec=3)
1.000000000000... - 6.380956565426...*q - 23.18584547617...*q^2 + O(q^3)

sage: x,y = var("x,y")
sage: WeakModularFormsRing()((x+1)/(x^3-y^2)).q_expansion_fixed_d(prec=2, fix_prec = True)
1/864*q^-1 + O(1)
sage: WeakModularFormsRing()((x+1)/(x^3-y^2)).q_expansion_fixed_d(prec=2)
1/864*q^-1 + 1/6 + 119/24*q + O(q^2)

sage: j_inv = WeakModularFormsRing(n=infinity, red_hom=True).j_inv()
sage: j_inv.q_expansion_fixed_d(prec=3)
q^-1 + 24 + 276*q + 2048*q^2 + O(q^3)

sage: E2 = QuasiModularFormsRing(n=infinity, red_hom=True).E2()
sage: E2.q_expansion_fixed_d(prec=3)
1 - 8*q - 8*q^2 + O(q^3)

sage: x,y = var("x,y")
sage: WeakModularFormsRing(n=infinity)((x+1)/(x-y^2)).q_expansion_fixed_d(prec=2, fix_prec = True)
1/32*q^-1 + O(1)
sage: WeakModularFormsRing(n=infinity)((x+1)/(x-y^2)).q_expansion_fixed_d(prec=2)
1/32*q^-1 + 1/2 + 39/8*q + O(q^2)

sage: (WeakModularFormsRing(n=14).J_inv()^3).q_expansion_fixed_d(prec=2)
2.933373093...e-6*q^-3 + 0.0002320999814...*q^-2 + 0.009013529265...*q^-1 + 0.2292916854... + 4.303583833...*q + O(q^2)

Python

>>> from sage.all import *
>>> from sage.modular.modform_hecketriangle.graded_ring import WeakModularFormsRing, QuasiModularFormsRing
>>> j_inv = WeakModularFormsRing(red_hom=True).j_inv()
>>> j_inv.q_expansion_fixed_d(prec=Integer(3))
q^-1 + 744 + 196884*q + 21493760*q^2 + O(q^3)

>>> E2 = QuasiModularFormsRing(n=Integer(5), red_hom=True).E2()
>>> E2.q_expansion_fixed_d(prec=Integer(3))
1.000000000000... - 6.380956565426...*q - 23.18584547617...*q^2 + O(q^3)

>>> x,y = var("x,y")
>>> WeakModularFormsRing()((x+Integer(1))/(x**Integer(3)-y**Integer(2))).q_expansion_fixed_d(prec=Integer(2), fix_prec = True)
1/864*q^-1 + O(1)
>>> WeakModularFormsRing()((x+Integer(1))/(x**Integer(3)-y**Integer(2))).q_expansion_fixed_d(prec=Integer(2))
1/864*q^-1 + 1/6 + 119/24*q + O(q^2)

>>> j_inv = WeakModularFormsRing(n=infinity, red_hom=True).j_inv()
>>> j_inv.q_expansion_fixed_d(prec=Integer(3))
q^-1 + 24 + 276*q + 2048*q^2 + O(q^3)

>>> E2 = QuasiModularFormsRing(n=infinity, red_hom=True).E2()
>>> E2.q_expansion_fixed_d(prec=Integer(3))
1 - 8*q - 8*q^2 + O(q^3)

>>> x,y = var("x,y")
>>> WeakModularFormsRing(n=infinity)((x+Integer(1))/(x-y**Integer(2))).q_expansion_fixed_d(prec=Integer(2), fix_prec = True)
1/32*q^-1 + O(1)
>>> WeakModularFormsRing(n=infinity)((x+Integer(1))/(x-y**Integer(2))).q_expansion_fixed_d(prec=Integer(2))
1/32*q^-1 + 1/2 + 39/8*q + O(q^2)

>>> (WeakModularFormsRing(n=Integer(14)).J_inv()**Integer(3)).q_expansion_fixed_d(prec=Integer(2))
2.933373093...e-6*q^-3 + 0.0002320999814...*q^-2 + 0.009013529265...*q^-1 + 0.2292916854... + 4.303583833...*q + O(q^2)

Sage Live

from sage.modular.modform_hecketriangle.graded_ring import WeakModularFormsRing, QuasiModularFormsRing
j_inv = WeakModularFormsRing(red_hom=True).j_inv()
j_inv.q_expansion_fixed_d(prec=3)
E2 = QuasiModularFormsRing(n=5, red_hom=True).E2()
E2.q_expansion_fixed_d(prec=3)
x,y = var("x,y")
WeakModularFormsRing()((x+1)/(x^3-y^2)).q_expansion_fixed_d(prec=2, fix_prec = True)
WeakModularFormsRing()((x+1)/(x^3-y^2)).q_expansion_fixed_d(prec=2)
j_inv = WeakModularFormsRing(n=infinity, red_hom=True).j_inv()
j_inv.q_expansion_fixed_d(prec=3)
E2 = QuasiModularFormsRing(n=infinity, red_hom=True).E2()
E2.q_expansion_fixed_d(prec=3)
x,y = var("x,y")
WeakModularFormsRing(n=infinity)((x+1)/(x-y^2)).q_expansion_fixed_d(prec=2, fix_prec = True)
WeakModularFormsRing(n=infinity)((x+1)/(x-y^2)).q_expansion_fixed_d(prec=2)
(WeakModularFormsRing(n=14).J_inv()^3).q_expansion_fixed_d(prec=2)

q_expansion_vector(min_exp=None, max_exp=None, prec=None, **kwargs)[source]¶

Return (part of) the Laurent series expansion of self as a vector.

INPUT:

min_exp – integer specifying the first coefficient to be used for the vector. Default: None, meaning that the first non-trivial coefficient is used.
max_exp – integer specifying the last coefficient to be used for the vector. Default: None, meaning that the default precision + 1 is used.
prec – integer specifying the precision of the underlying Laurent series. Default: None, meaning that max_exp + 1 is used.

OUTPUT:

A vector of size max_exp - min_exp over the coefficient ring of self, determined by the corresponding Laurent series coefficients.

EXAMPLES:

Sage

sage: from sage.modular.modform_hecketriangle.graded_ring import WeakModularFormsRing
sage: f = WeakModularFormsRing(red_hom=True).j_inv()^3
sage: f.q_expansion(prec=3)
q^-3 + 31/(24*d)*q^-2 + 20845/(27648*d^2)*q^-1 + 7058345/(26873856*d^3) + 30098784355/(495338913792*d^4)*q + 175372747465/(17832200896512*d^5)*q^2 + O(q^3)
sage: v = f.q_expansion_vector(max_exp=1, prec=3)
sage: v
(1, 31/(24*d), 20845/(27648*d^2), 7058345/(26873856*d^3), 30098784355/(495338913792*d^4))
sage: v.parent()
Vector space of dimension 5 over Fraction Field of Univariate Polynomial Ring in d over Integer Ring
sage: f.q_expansion_vector(min_exp=1, max_exp=2)
(30098784355/(495338913792*d^4), 175372747465/(17832200896512*d^5))
sage: f.q_expansion_vector(min_exp=1, max_exp=2, fix_d=True)
(541778118390, 151522053809760)

sage: f = WeakModularFormsRing(n=infinity, red_hom=True).j_inv()^3
sage: f.q_expansion_fixed_d(prec=3)
q^-3 + 72*q^-2 + 2556*q^-1 + 59712 + 1033974*q + 14175648*q^2 + O(q^3)
sage: v = f.q_expansion_vector(max_exp=1, prec=3, fix_d=True)
sage: v
(1, 72, 2556, 59712, 1033974)
sage: v.parent()
Vector space of dimension 5 over Rational Field
sage: f.q_expansion_vector(min_exp=1, max_exp=2)
(516987/(8388608*d^4), 442989/(33554432*d^5))

Python

>>> from sage.all import *
>>> from sage.modular.modform_hecketriangle.graded_ring import WeakModularFormsRing
>>> f = WeakModularFormsRing(red_hom=True).j_inv()**Integer(3)
>>> f.q_expansion(prec=Integer(3))
q^-3 + 31/(24*d)*q^-2 + 20845/(27648*d^2)*q^-1 + 7058345/(26873856*d^3) + 30098784355/(495338913792*d^4)*q + 175372747465/(17832200896512*d^5)*q^2 + O(q^3)
>>> v = f.q_expansion_vector(max_exp=Integer(1), prec=Integer(3))
>>> v
(1, 31/(24*d), 20845/(27648*d^2), 7058345/(26873856*d^3), 30098784355/(495338913792*d^4))
>>> v.parent()
Vector space of dimension 5 over Fraction Field of Univariate Polynomial Ring in d over Integer Ring
>>> f.q_expansion_vector(min_exp=Integer(1), max_exp=Integer(2))
(30098784355/(495338913792*d^4), 175372747465/(17832200896512*d^5))
>>> f.q_expansion_vector(min_exp=Integer(1), max_exp=Integer(2), fix_d=True)
(541778118390, 151522053809760)

>>> f = WeakModularFormsRing(n=infinity, red_hom=True).j_inv()**Integer(3)
>>> f.q_expansion_fixed_d(prec=Integer(3))
q^-3 + 72*q^-2 + 2556*q^-1 + 59712 + 1033974*q + 14175648*q^2 + O(q^3)
>>> v = f.q_expansion_vector(max_exp=Integer(1), prec=Integer(3), fix_d=True)
>>> v
(1, 72, 2556, 59712, 1033974)
>>> v.parent()
Vector space of dimension 5 over Rational Field
>>> f.q_expansion_vector(min_exp=Integer(1), max_exp=Integer(2))
(516987/(8388608*d^4), 442989/(33554432*d^5))

Sage Live

from sage.modular.modform_hecketriangle.graded_ring import WeakModularFormsRing
f = WeakModularFormsRing(red_hom=True).j_inv()^3
f.q_expansion(prec=3)
v = f.q_expansion_vector(max_exp=1, prec=3)
v
v.parent()
f.q_expansion_vector(min_exp=1, max_exp=2)
f.q_expansion_vector(min_exp=1, max_exp=2, fix_d=True)
f = WeakModularFormsRing(n=infinity, red_hom=True).j_inv()^3
f.q_expansion_fixed_d(prec=3)
v = f.q_expansion_vector(max_exp=1, prec=3, fix_d=True)
v
v.parent()
f.q_expansion_vector(min_exp=1, max_exp=2)

rat()[source]¶

Return the rational function representing self.

EXAMPLES:

Sage

sage: from sage.modular.modform_hecketriangle.graded_ring import ModularFormsRing
sage: ModularFormsRing(n=12).Delta().rat()
x^30*d - x^18*y^2*d

Python

>>> from sage.all import *
>>> from sage.modular.modform_hecketriangle.graded_ring import ModularFormsRing
>>> ModularFormsRing(n=Integer(12)).Delta().rat()
x^30*d - x^18*y^2*d

Sage Live

from sage.modular.modform_hecketriangle.graded_ring import ModularFormsRing
ModularFormsRing(n=12).Delta().rat()

reduce(force=False)[source]¶

In case self.parent().has_reduce_hom() == True (or force==True) and self is homogeneous the converted element lying in the corresponding homogeneous_part is returned.

Otherwise self is returned.

EXAMPLES:

Sage

sage: from sage.modular.modform_hecketriangle.graded_ring import ModularFormsRing
sage: E2 = ModularFormsRing(n=7).E2().reduce()
sage: E2.parent()
QuasiModularFormsRing(n=7) over Integer Ring
sage: E2 = ModularFormsRing(n=7, red_hom=True).E2().reduce()
sage: E2.parent()
QuasiModularForms(n=7, k=2, ep=-1) over Integer Ring
sage: ModularFormsRing(n=7)(x+1).reduce().parent()
ModularFormsRing(n=7) over Integer Ring
sage: E2 = ModularFormsRing(n=7).E2().reduce(force=True)
sage: E2.parent()
QuasiModularForms(n=7, k=2, ep=-1) over Integer Ring
sage: ModularFormsRing(n=7)(x+1).reduce(force=True).parent()
ModularFormsRing(n=7) over Integer Ring

sage: y = var("y")
sage: ModularFormsRing(n=infinity)(x-y^2).reduce(force=True)
64*q - 512*q^2 + 1792*q^3 - 4096*q^4 + O(q^5)

Python

>>> from sage.all import *
>>> from sage.modular.modform_hecketriangle.graded_ring import ModularFormsRing
>>> E2 = ModularFormsRing(n=Integer(7)).E2().reduce()
>>> E2.parent()
QuasiModularFormsRing(n=7) over Integer Ring
>>> E2 = ModularFormsRing(n=Integer(7), red_hom=True).E2().reduce()
>>> E2.parent()
QuasiModularForms(n=7, k=2, ep=-1) over Integer Ring
>>> ModularFormsRing(n=Integer(7))(x+Integer(1)).reduce().parent()
ModularFormsRing(n=7) over Integer Ring
>>> E2 = ModularFormsRing(n=Integer(7)).E2().reduce(force=True)
>>> E2.parent()
QuasiModularForms(n=7, k=2, ep=-1) over Integer Ring
>>> ModularFormsRing(n=Integer(7))(x+Integer(1)).reduce(force=True).parent()
ModularFormsRing(n=7) over Integer Ring

>>> y = var("y")
>>> ModularFormsRing(n=infinity)(x-y**Integer(2)).reduce(force=True)
64*q - 512*q^2 + 1792*q^3 - 4096*q^4 + O(q^5)

Sage Live

from sage.modular.modform_hecketriangle.graded_ring import ModularFormsRing
E2 = ModularFormsRing(n=7).E2().reduce()
E2.parent()
E2 = ModularFormsRing(n=7, red_hom=True).E2().reduce()
E2.parent()
ModularFormsRing(n=7)(x+1).reduce().parent()
E2 = ModularFormsRing(n=7).E2().reduce(force=True)
E2.parent()
ModularFormsRing(n=7)(x+1).reduce(force=True).parent()
y = var("y")
ModularFormsRing(n=infinity)(x-y^2).reduce(force=True)

reduced_parent()[source]¶

Return the space with the analytic type of self. If self is homogeneous the corresponding FormsSpace is returned.

I.e. return the smallest known ambient space of self.

EXAMPLES:

Sage

sage: from sage.modular.modform_hecketriangle.graded_ring import QuasiMeromorphicModularFormsRing
sage: Delta = QuasiMeromorphicModularFormsRing(n=7).Delta()
sage: Delta.parent()
QuasiMeromorphicModularFormsRing(n=7) over Integer Ring
sage: Delta.reduced_parent()
CuspForms(n=7, k=12, ep=1) over Integer Ring
sage: el = QuasiMeromorphicModularFormsRing()(x+1)
sage: el.parent()
QuasiMeromorphicModularFormsRing(n=3) over Integer Ring
sage: el.reduced_parent()
ModularFormsRing(n=3) over Integer Ring

sage: y = var("y")
sage: QuasiMeromorphicModularFormsRing(n=infinity)(x-y^2).reduced_parent()
ModularForms(n=+Infinity, k=4, ep=1) over Integer Ring
sage: QuasiMeromorphicModularFormsRing(n=infinity)(x*(x-y^2)).reduced_parent()
CuspForms(n=+Infinity, k=8, ep=1) over Integer Ring

Python

>>> from sage.all import *
>>> from sage.modular.modform_hecketriangle.graded_ring import QuasiMeromorphicModularFormsRing
>>> Delta = QuasiMeromorphicModularFormsRing(n=Integer(7)).Delta()
>>> Delta.parent()
QuasiMeromorphicModularFormsRing(n=7) over Integer Ring
>>> Delta.reduced_parent()
CuspForms(n=7, k=12, ep=1) over Integer Ring
>>> el = QuasiMeromorphicModularFormsRing()(x+Integer(1))
>>> el.parent()
QuasiMeromorphicModularFormsRing(n=3) over Integer Ring
>>> el.reduced_parent()
ModularFormsRing(n=3) over Integer Ring

>>> y = var("y")
>>> QuasiMeromorphicModularFormsRing(n=infinity)(x-y**Integer(2)).reduced_parent()
ModularForms(n=+Infinity, k=4, ep=1) over Integer Ring
>>> QuasiMeromorphicModularFormsRing(n=infinity)(x*(x-y**Integer(2))).reduced_parent()
CuspForms(n=+Infinity, k=8, ep=1) over Integer Ring

Sage Live

from sage.modular.modform_hecketriangle.graded_ring import QuasiMeromorphicModularFormsRing
Delta = QuasiMeromorphicModularFormsRing(n=7).Delta()
Delta.parent()
Delta.reduced_parent()
el = QuasiMeromorphicModularFormsRing()(x+1)
el.parent()
el.reduced_parent()
y = var("y")
QuasiMeromorphicModularFormsRing(n=infinity)(x-y^2).reduced_parent()
QuasiMeromorphicModularFormsRing(n=infinity)(x*(x-y^2)).reduced_parent()

serre_derivative()[source]¶

Return the Serre derivative of self.

Note that the parent might (probably will) change. However a modular element is returned if self was already modular.

If parent.has_reduce_hom() == True then the result is reduced to be an element of the corresponding forms space if possible.

In particular this is the case if self is a (homogeneous) element of a forms space.

EXAMPLES:

Sage

sage: from sage.modular.modform_hecketriangle.graded_ring import QuasiMeromorphicModularFormsRing
sage: MR = QuasiMeromorphicModularFormsRing(n=7, red_hom=True)
sage: n = MR.hecke_n()
sage: Delta = MR.Delta().full_reduce()
sage: E2 = MR.E2().full_reduce()
sage: E4 = MR.E4().full_reduce()
sage: E6 = MR.E6().full_reduce()
sage: f_rho = MR.f_rho().full_reduce()
sage: f_i = MR.f_i().full_reduce()
sage: f_inf = MR.f_inf().full_reduce()

sage: f_rho.serre_derivative() == -1/n * f_i
True
sage: f_i.serre_derivative()   == -1/2 * E4 * f_rho
True
sage: f_inf.serre_derivative() == 0
True
sage: E2.serre_derivative()    == -(n-2)/(4*n) * (E2^2 + E4)
True
sage: E4.serre_derivative()    == -(n-2)/n * E6
True
sage: E6.serre_derivative()    == -1/2 * E4^2 - (n-3)/n * E6^2 / E4
True
sage: E6.serre_derivative().parent()
ModularForms(n=7, k=8, ep=1) over Integer Ring

sage: MR = QuasiMeromorphicModularFormsRing(n=infinity, red_hom=True)
sage: Delta = MR.Delta().full_reduce()
sage: E2 = MR.E2().full_reduce()
sage: E4 = MR.E4().full_reduce()
sage: E6 = MR.E6().full_reduce()
sage: f_i = MR.f_i().full_reduce()
sage: f_inf = MR.f_inf().full_reduce()

sage: E4.serre_derivative()    == -E4 * f_i
True
sage: f_i.serre_derivative()   == -1/2 * E4
True
sage: f_inf.serre_derivative() == 0
True
sage: E2.serre_derivative()    == -1/4 * (E2^2 + E4)
True
sage: E4.serre_derivative()    == -E6
True
sage: E6.serre_derivative()    == -1/2 * E4^2 - E6^2 / E4
True
sage: E6.serre_derivative().parent()
ModularForms(n=+Infinity, k=8, ep=1) over Integer Ring

Python

>>> from sage.all import *
>>> from sage.modular.modform_hecketriangle.graded_ring import QuasiMeromorphicModularFormsRing
>>> MR = QuasiMeromorphicModularFormsRing(n=Integer(7), red_hom=True)
>>> n = MR.hecke_n()
>>> Delta = MR.Delta().full_reduce()
>>> E2 = MR.E2().full_reduce()
>>> E4 = MR.E4().full_reduce()
>>> E6 = MR.E6().full_reduce()
>>> f_rho = MR.f_rho().full_reduce()
>>> f_i = MR.f_i().full_reduce()
>>> f_inf = MR.f_inf().full_reduce()

>>> f_rho.serre_derivative() == -Integer(1)/n * f_i
True
>>> f_i.serre_derivative()   == -Integer(1)/Integer(2) * E4 * f_rho
True
>>> f_inf.serre_derivative() == Integer(0)
True
>>> E2.serre_derivative()    == -(n-Integer(2))/(Integer(4)*n) * (E2**Integer(2) + E4)
True
>>> E4.serre_derivative()    == -(n-Integer(2))/n * E6
True
>>> E6.serre_derivative()    == -Integer(1)/Integer(2) * E4**Integer(2) - (n-Integer(3))/n * E6**Integer(2) / E4
True
>>> E6.serre_derivative().parent()
ModularForms(n=7, k=8, ep=1) over Integer Ring

>>> MR = QuasiMeromorphicModularFormsRing(n=infinity, red_hom=True)
>>> Delta = MR.Delta().full_reduce()
>>> E2 = MR.E2().full_reduce()
>>> E4 = MR.E4().full_reduce()
>>> E6 = MR.E6().full_reduce()
>>> f_i = MR.f_i().full_reduce()
>>> f_inf = MR.f_inf().full_reduce()

>>> E4.serre_derivative()    == -E4 * f_i
True
>>> f_i.serre_derivative()   == -Integer(1)/Integer(2) * E4
True
>>> f_inf.serre_derivative() == Integer(0)
True
>>> E2.serre_derivative()    == -Integer(1)/Integer(4) * (E2**Integer(2) + E4)
True
>>> E4.serre_derivative()    == -E6
True
>>> E6.serre_derivative()    == -Integer(1)/Integer(2) * E4**Integer(2) - E6**Integer(2) / E4
True
>>> E6.serre_derivative().parent()
ModularForms(n=+Infinity, k=8, ep=1) over Integer Ring

Sage Live

from sage.modular.modform_hecketriangle.graded_ring import QuasiMeromorphicModularFormsRing
MR = QuasiMeromorphicModularFormsRing(n=7, red_hom=True)
n = MR.hecke_n()
Delta = MR.Delta().full_reduce()
E2 = MR.E2().full_reduce()
E4 = MR.E4().full_reduce()
E6 = MR.E6().full_reduce()
f_rho = MR.f_rho().full_reduce()
f_i = MR.f_i().full_reduce()
f_inf = MR.f_inf().full_reduce()
f_rho.serre_derivative() == -1/n * f_i
f_i.serre_derivative()   == -1/2 * E4 * f_rho
f_inf.serre_derivative() == 0
E2.serre_derivative()    == -(n-2)/(4*n) * (E2^2 + E4)
E4.serre_derivative()    == -(n-2)/n * E6
E6.serre_derivative()    == -1/2 * E4^2 - (n-3)/n * E6^2 / E4
E6.serre_derivative().parent()
MR = QuasiMeromorphicModularFormsRing(n=infinity, red_hom=True)
Delta = MR.Delta().full_reduce()
E2 = MR.E2().full_reduce()
E4 = MR.E4().full_reduce()
E6 = MR.E6().full_reduce()
f_i = MR.f_i().full_reduce()
f_inf = MR.f_inf().full_reduce()
E4.serre_derivative()    == -E4 * f_i
f_i.serre_derivative()   == -1/2 * E4
f_inf.serre_derivative() == 0
E2.serre_derivative()    == -1/4 * (E2^2 + E4)
E4.serre_derivative()    == -E6
E6.serre_derivative()    == -1/2 * E4^2 - E6^2 / E4
E6.serre_derivative().parent()

sqrt()[source]¶

Return the square root of self if it exists.

I.e. the element corresponding to sqrt(self.rat()).

Whether this works or not depends on whether sqrt(self.rat()) works and coerces into self.parent().rat_field().

Note that the parent might (probably will) change.

If parent.has_reduce_hom() == True then the result is reduced to be an element of the corresponding forms space if possible.

In particular this is the case if self is a (homogeneous) element of a forms space.

EXAMPLES:

Sage

sage: from sage.modular.modform_hecketriangle.space import QuasiModularForms
sage: E2 = QuasiModularForms(k=2, ep=-1).E2()
sage: (E2^2).sqrt()
1 - 24*q - 72*q^2 - 96*q^3 - 168*q^4 + O(q^5)
sage: (E2^2).sqrt() == E2
True

Python

>>> from sage.all import *
>>> from sage.modular.modform_hecketriangle.space import QuasiModularForms
>>> E2 = QuasiModularForms(k=Integer(2), ep=-Integer(1)).E2()
>>> (E2**Integer(2)).sqrt()
1 - 24*q - 72*q^2 - 96*q^3 - 168*q^4 + O(q^5)
>>> (E2**Integer(2)).sqrt() == E2
True

Sage Live

from sage.modular.modform_hecketriangle.space import QuasiModularForms
E2 = QuasiModularForms(k=2, ep=-1).E2()
(E2^2).sqrt()
(E2^2).sqrt() == E2

weight()[source]¶

Return the weight of self.

EXAMPLES:

Sage

sage: from sage.modular.modform_hecketriangle.graded_ring import QuasiModularFormsRing
sage: from sage.modular.modform_hecketriangle.space import ModularForms
sage: x, y, z, d = var("x,y,z,d")                                           # needs sage.symbolic
sage: QuasiModularFormsRing()(x+y).weight() is None                         # needs sage.symbolic
True
sage: ModularForms(n=18).f_i().weight()
9/4
sage: ModularForms(n=infinity).f_inf().weight()
4

Python

>>> from sage.all import *
>>> from sage.modular.modform_hecketriangle.graded_ring import QuasiModularFormsRing
>>> from sage.modular.modform_hecketriangle.space import ModularForms
>>> x, y, z, d = var("x,y,z,d")                                           # needs sage.symbolic
>>> QuasiModularFormsRing()(x+y).weight() is None                         # needs sage.symbolic
True
>>> ModularForms(n=Integer(18)).f_i().weight()
9/4
>>> ModularForms(n=infinity).f_inf().weight()
4

Sage Live

from sage.modular.modform_hecketriangle.graded_ring import QuasiModularFormsRing
from sage.modular.modform_hecketriangle.space import ModularForms
x, y, z, d = var("x,y,z,d")                                           # needs sage.symbolic
QuasiModularFormsRing()(x+y).weight() is None                         # needs sage.symbolic
ModularForms(n=18).f_i().weight()
ModularForms(n=infinity).f_inf().weight()