Subspaces of modular forms for Hecke triangle groups

AUTHORS:

  • Jonas Jermann (2013): initial version

sage.modular.modform_hecketriangle.subspace.ModularFormsSubSpace(*args, **kwargs)[source]

Create a modular forms subspace generated by the supplied arguments if possible. Instead of a list of generators also multiple input arguments can be used. If reduce=True then the corresponding ambient space is chosen as small as possible. If no subspace is available then the ambient space is returned.

EXAMPLES:

sage: from sage.modular.modform_hecketriangle.subspace import ModularFormsSubSpace
sage: from sage.modular.modform_hecketriangle.space import ModularForms
sage: MF = ModularForms()
sage: subspace = ModularFormsSubSpace(MF.E4()^3, MF.E6()^2+MF.Delta(), MF.Delta())
sage: subspace
Subspace of dimension 2 of ModularForms(n=3, k=12, ep=1) over Integer Ring
sage: subspace.ambient_space()
ModularForms(n=3, k=12, ep=1) over Integer Ring
sage: subspace.gens()
[1 + 720*q + 179280*q^2 + 16954560*q^3 + 396974160*q^4 + O(q^5), 1 - 1007*q + 220728*q^2 + 16519356*q^3 + 399516304*q^4 + O(q^5)]
sage: ModularFormsSubSpace(MF.E4()^3-MF.E6()^2, reduce=True).ambient_space()
CuspForms(n=3, k=12, ep=1) over Integer Ring
sage: ModularFormsSubSpace(MF.E4()^3-MF.E6()^2, MF.J_inv()*MF.E4()^3, reduce=True)
WeakModularForms(n=3, k=12, ep=1) over Integer Ring

>>> from sage.all import *
>>> from sage.modular.modform_hecketriangle.subspace import ModularFormsSubSpace
>>> from sage.modular.modform_hecketriangle.space import ModularForms
>>> MF = ModularForms()
>>> subspace = ModularFormsSubSpace(MF.E4()**Integer(3), MF.E6()**Integer(2)+MF.Delta(), MF.Delta())
>>> subspace
Subspace of dimension 2 of ModularForms(n=3, k=12, ep=1) over Integer Ring
>>> subspace.ambient_space()
ModularForms(n=3, k=12, ep=1) over Integer Ring
>>> subspace.gens()
[1 + 720*q + 179280*q^2 + 16954560*q^3 + 396974160*q^4 + O(q^5), 1 - 1007*q + 220728*q^2 + 16519356*q^3 + 399516304*q^4 + O(q^5)]
>>> ModularFormsSubSpace(MF.E4()**Integer(3)-MF.E6()**Integer(2), reduce=True).ambient_space()
CuspForms(n=3, k=12, ep=1) over Integer Ring
>>> ModularFormsSubSpace(MF.E4()**Integer(3)-MF.E6()**Integer(2), MF.J_inv()*MF.E4()**Integer(3), reduce=True)
WeakModularForms(n=3, k=12, ep=1) over Integer Ring

from sage.modular.modform_hecketriangle.subspace import ModularFormsSubSpace
from sage.modular.modform_hecketriangle.space import ModularForms
MF = ModularForms()
subspace = ModularFormsSubSpace(MF.E4()^3, MF.E6()^2+MF.Delta(), MF.Delta())
subspace
subspace.ambient_space()
subspace.gens()
ModularFormsSubSpace(MF.E4()^3-MF.E6()^2, reduce=True).ambient_space()
ModularFormsSubSpace(MF.E4()^3-MF.E6()^2, MF.J_inv()*MF.E4()^3, reduce=True)
class sage.modular.modform_hecketriangle.subspace.SubSpaceForms(ambient_space, basis, check)[source]

Bases: FormsSpace_abstract, Module, UniqueRepresentation

Submodule of (Hecke) forms in the given ambient space for the given basis.

basis()[source]

Return the basis of self in the ambient space.

EXAMPLES:

sage: from sage.modular.modform_hecketriangle.space import ModularForms
sage: MF = ModularForms(n=6, k=20, ep=1)
sage: subspace = MF.subspace([(MF.Delta()*MF.E4()^2).as_ring_element(), MF.gen(0)])
sage: subspace.basis()
[q + 78*q^2 + 2781*q^3 + 59812*q^4 + O(q^5), 1 + 360360*q^4 + O(q^5)]
sage: subspace.basis()[0].parent() == MF
True

>>> from sage.all import *
>>> from sage.modular.modform_hecketriangle.space import ModularForms
>>> MF = ModularForms(n=Integer(6), k=Integer(20), ep=Integer(1))
>>> subspace = MF.subspace([(MF.Delta()*MF.E4()**Integer(2)).as_ring_element(), MF.gen(Integer(0))])
>>> subspace.basis()
[q + 78*q^2 + 2781*q^3 + 59812*q^4 + O(q^5), 1 + 360360*q^4 + O(q^5)]
>>> subspace.basis()[Integer(0)].parent() == MF
True

from sage.modular.modform_hecketriangle.space import ModularForms
MF = ModularForms(n=6, k=20, ep=1)
subspace = MF.subspace([(MF.Delta()*MF.E4()^2).as_ring_element(), MF.gen(0)])
subspace.basis()
subspace.basis()[0].parent() == MF
change_ambient_space(new_ambient_space)[source]

Return a new subspace with the same basis but inside a different ambient space (if possible).

EXAMPLES:

sage: from sage.modular.modform_hecketriangle.space import ModularForms, QuasiModularForms
sage: MF = ModularForms(n=6, k=20, ep=1)
sage: subspace = MF.subspace([MF.Delta()*MF.E4()^2, MF.gen(0)])
sage: new_ambient_space = QuasiModularForms(n=6, k=20, ep=1)
sage: subspace.change_ambient_space(new_ambient_space)    # long time
Subspace of dimension 2 of QuasiModularForms(n=6, k=20, ep=1) over Integer Ring

>>> from sage.all import *
>>> from sage.modular.modform_hecketriangle.space import ModularForms, QuasiModularForms
>>> MF = ModularForms(n=Integer(6), k=Integer(20), ep=Integer(1))
>>> subspace = MF.subspace([MF.Delta()*MF.E4()**Integer(2), MF.gen(Integer(0))])
>>> new_ambient_space = QuasiModularForms(n=Integer(6), k=Integer(20), ep=Integer(1))
>>> subspace.change_ambient_space(new_ambient_space)    # long time
Subspace of dimension 2 of QuasiModularForms(n=6, k=20, ep=1) over Integer Ring

from sage.modular.modform_hecketriangle.space import ModularForms, QuasiModularForms
MF = ModularForms(n=6, k=20, ep=1)
subspace = MF.subspace([MF.Delta()*MF.E4()^2, MF.gen(0)])
new_ambient_space = QuasiModularForms(n=6, k=20, ep=1)
subspace.change_ambient_space(new_ambient_space)    # long time
change_ring(new_base_ring)[source]

Return the same space as self but over a new base ring new_base_ring.

EXAMPLES:

sage: from sage.modular.modform_hecketriangle.space import ModularForms
sage: MF = ModularForms(n=6, k=20, ep=1)
sage: subspace = MF.subspace([MF.Delta()*MF.E4()^2, MF.gen(0)])
sage: subspace.change_ring(QQ)
Subspace of dimension 2 of ModularForms(n=6, k=20, ep=1) over Rational Field
sage: subspace.change_ring(CC)
Traceback (most recent call last):
...
NotImplementedError

>>> from sage.all import *
>>> from sage.modular.modform_hecketriangle.space import ModularForms
>>> MF = ModularForms(n=Integer(6), k=Integer(20), ep=Integer(1))
>>> subspace = MF.subspace([MF.Delta()*MF.E4()**Integer(2), MF.gen(Integer(0))])
>>> subspace.change_ring(QQ)
Subspace of dimension 2 of ModularForms(n=6, k=20, ep=1) over Rational Field
>>> subspace.change_ring(CC)
Traceback (most recent call last):
...
NotImplementedError

from sage.modular.modform_hecketriangle.space import ModularForms
MF = ModularForms(n=6, k=20, ep=1)
subspace = MF.subspace([MF.Delta()*MF.E4()^2, MF.gen(0)])
subspace.change_ring(QQ)
subspace.change_ring(CC)
contains_coeff_ring()[source]

Return whether self contains its coefficient ring.

EXAMPLES:

sage: from sage.modular.modform_hecketriangle.space import ModularForms
sage: MF = ModularForms(k=0, ep=1, n=8)
sage: subspace = MF.subspace([1])
sage: subspace.contains_coeff_ring()
True
sage: subspace = MF.subspace([])
sage: subspace.contains_coeff_ring()
False
sage: MF = ModularForms(k=0, ep=-1, n=8)
sage: subspace = MF.subspace([])
sage: subspace.contains_coeff_ring()
False

>>> from sage.all import *
>>> from sage.modular.modform_hecketriangle.space import ModularForms
>>> MF = ModularForms(k=Integer(0), ep=Integer(1), n=Integer(8))
>>> subspace = MF.subspace([Integer(1)])
>>> subspace.contains_coeff_ring()
True
>>> subspace = MF.subspace([])
>>> subspace.contains_coeff_ring()
False
>>> MF = ModularForms(k=Integer(0), ep=-Integer(1), n=Integer(8))
>>> subspace = MF.subspace([])
>>> subspace.contains_coeff_ring()
False

from sage.modular.modform_hecketriangle.space import ModularForms
MF = ModularForms(k=0, ep=1, n=8)
subspace = MF.subspace([1])
subspace.contains_coeff_ring()
subspace = MF.subspace([])
subspace.contains_coeff_ring()
MF = ModularForms(k=0, ep=-1, n=8)
subspace = MF.subspace([])
subspace.contains_coeff_ring()
coordinate_vector(v)[source]

Return the coordinate vector of v with respect to the basis self.gens().

INPUT:

  • v – an element of self

OUTPUT:

The coordinate vector of v with respect to the basis self.gens().

Note: The coordinate vector is not an element of self.module().

EXAMPLES:

sage: from sage.modular.modform_hecketriangle.space import ModularForms, QuasiCuspForms
sage: MF = ModularForms(n=6, k=20, ep=1)
sage: subspace = MF.subspace([(MF.Delta()*MF.E4()^2).as_ring_element(), MF.gen(0)])
sage: subspace.coordinate_vector(MF.gen(0) + MF.Delta()*MF.E4()^2).parent()
Vector space of dimension 2 over Fraction Field of Univariate Polynomial Ring in d over Integer Ring
sage: subspace.coordinate_vector(MF.gen(0) + MF.Delta()*MF.E4()^2)
(1, 1)

sage: MF = ModularForms(n=4, k=24, ep=-1)
sage: subspace = MF.subspace([MF.gen(0), MF.gen(2)])
sage: subspace.coordinate_vector(subspace.gen(0)).parent()
Vector space of dimension 2 over Fraction Field of Univariate Polynomial Ring in d over Integer Ring
sage: subspace.coordinate_vector(subspace.gen(0))
(1, 0)

sage: MF = QuasiCuspForms(n=infinity, k=12, ep=1)
sage: subspace = MF.subspace([MF.Delta(), MF.E4()*MF.f_inf()*MF.E2()*MF.f_i(), MF.E4()*MF.f_inf()*MF.E2()^2, MF.E4()*MF.f_inf()*(MF.E4()-MF.E2()^2)])
sage: el = MF.E4()*MF.f_inf()*(7*MF.E4() - 3*MF.E2()^2)
sage: subspace.coordinate_vector(el)
(7, 0, -3)
sage: subspace.ambient_coordinate_vector(el)
(7, 21/(8*d), 0, -3)

>>> from sage.all import *
>>> from sage.modular.modform_hecketriangle.space import ModularForms, QuasiCuspForms
>>> MF = ModularForms(n=Integer(6), k=Integer(20), ep=Integer(1))
>>> subspace = MF.subspace([(MF.Delta()*MF.E4()**Integer(2)).as_ring_element(), MF.gen(Integer(0))])
>>> subspace.coordinate_vector(MF.gen(Integer(0)) + MF.Delta()*MF.E4()**Integer(2)).parent()
Vector space of dimension 2 over Fraction Field of Univariate Polynomial Ring in d over Integer Ring
>>> subspace.coordinate_vector(MF.gen(Integer(0)) + MF.Delta()*MF.E4()**Integer(2))
(1, 1)

>>> MF = ModularForms(n=Integer(4), k=Integer(24), ep=-Integer(1))
>>> subspace = MF.subspace([MF.gen(Integer(0)), MF.gen(Integer(2))])
>>> subspace.coordinate_vector(subspace.gen(Integer(0))).parent()
Vector space of dimension 2 over Fraction Field of Univariate Polynomial Ring in d over Integer Ring
>>> subspace.coordinate_vector(subspace.gen(Integer(0)))
(1, 0)

>>> MF = QuasiCuspForms(n=infinity, k=Integer(12), ep=Integer(1))
>>> subspace = MF.subspace([MF.Delta(), MF.E4()*MF.f_inf()*MF.E2()*MF.f_i(), MF.E4()*MF.f_inf()*MF.E2()**Integer(2), MF.E4()*MF.f_inf()*(MF.E4()-MF.E2()**Integer(2))])
>>> el = MF.E4()*MF.f_inf()*(Integer(7)*MF.E4() - Integer(3)*MF.E2()**Integer(2))
>>> subspace.coordinate_vector(el)
(7, 0, -3)
>>> subspace.ambient_coordinate_vector(el)
(7, 21/(8*d), 0, -3)

from sage.modular.modform_hecketriangle.space import ModularForms, QuasiCuspForms
MF = ModularForms(n=6, k=20, ep=1)
subspace = MF.subspace([(MF.Delta()*MF.E4()^2).as_ring_element(), MF.gen(0)])
subspace.coordinate_vector(MF.gen(0) + MF.Delta()*MF.E4()^2).parent()
subspace.coordinate_vector(MF.gen(0) + MF.Delta()*MF.E4()^2)
MF = ModularForms(n=4, k=24, ep=-1)
subspace = MF.subspace([MF.gen(0), MF.gen(2)])
subspace.coordinate_vector(subspace.gen(0)).parent()
subspace.coordinate_vector(subspace.gen(0))
MF = QuasiCuspForms(n=infinity, k=12, ep=1)
subspace = MF.subspace([MF.Delta(), MF.E4()*MF.f_inf()*MF.E2()*MF.f_i(), MF.E4()*MF.f_inf()*MF.E2()^2, MF.E4()*MF.f_inf()*(MF.E4()-MF.E2()^2)])
el = MF.E4()*MF.f_inf()*(7*MF.E4() - 3*MF.E2()^2)
subspace.coordinate_vector(el)
subspace.ambient_coordinate_vector(el)
degree()[source]

Return the degree of self.

EXAMPLES:

sage: from sage.modular.modform_hecketriangle.space import ModularForms
sage: MF = ModularForms(n=6, k=20, ep=1)
sage: subspace = MF.subspace([(MF.Delta()*MF.E4()^2).as_ring_element(), MF.gen(0)])
sage: subspace.degree()
4
sage: subspace.degree() == subspace.ambient_space().degree()
True

>>> from sage.all import *
>>> from sage.modular.modform_hecketriangle.space import ModularForms
>>> MF = ModularForms(n=Integer(6), k=Integer(20), ep=Integer(1))
>>> subspace = MF.subspace([(MF.Delta()*MF.E4()**Integer(2)).as_ring_element(), MF.gen(Integer(0))])
>>> subspace.degree()
4
>>> subspace.degree() == subspace.ambient_space().degree()
True

from sage.modular.modform_hecketriangle.space import ModularForms
MF = ModularForms(n=6, k=20, ep=1)
subspace = MF.subspace([(MF.Delta()*MF.E4()^2).as_ring_element(), MF.gen(0)])
subspace.degree()
subspace.degree() == subspace.ambient_space().degree()
dimension()[source]

Return the dimension of self.

EXAMPLES:

sage: from sage.modular.modform_hecketriangle.space import ModularForms
sage: MF = ModularForms(n=6, k=20, ep=1)
sage: subspace = MF.subspace([(MF.Delta()*MF.E4()^2).as_ring_element(), MF.gen(0)])
sage: subspace.dimension()
2
sage: subspace.dimension() == len(subspace.gens())
True

>>> from sage.all import *
>>> from sage.modular.modform_hecketriangle.space import ModularForms
>>> MF = ModularForms(n=Integer(6), k=Integer(20), ep=Integer(1))
>>> subspace = MF.subspace([(MF.Delta()*MF.E4()**Integer(2)).as_ring_element(), MF.gen(Integer(0))])
>>> subspace.dimension()
2
>>> subspace.dimension() == len(subspace.gens())
True

from sage.modular.modform_hecketriangle.space import ModularForms
MF = ModularForms(n=6, k=20, ep=1)
subspace = MF.subspace([(MF.Delta()*MF.E4()^2).as_ring_element(), MF.gen(0)])
subspace.dimension()
subspace.dimension() == len(subspace.gens())
gens()[source]

Return the basis of self.

EXAMPLES:

sage: from sage.modular.modform_hecketriangle.space import ModularForms
sage: MF = ModularForms(n=6, k=20, ep=1)
sage: subspace = MF.subspace([(MF.Delta()*MF.E4()^2).as_ring_element(), MF.gen(0)])
sage: subspace.gens()
[q + 78*q^2 + 2781*q^3 + 59812*q^4 + O(q^5), 1 + 360360*q^4 + O(q^5)]
sage: subspace.gens()[0].parent() == subspace
True

>>> from sage.all import *
>>> from sage.modular.modform_hecketriangle.space import ModularForms
>>> MF = ModularForms(n=Integer(6), k=Integer(20), ep=Integer(1))
>>> subspace = MF.subspace([(MF.Delta()*MF.E4()**Integer(2)).as_ring_element(), MF.gen(Integer(0))])
>>> subspace.gens()
[q + 78*q^2 + 2781*q^3 + 59812*q^4 + O(q^5), 1 + 360360*q^4 + O(q^5)]
>>> subspace.gens()[Integer(0)].parent() == subspace
True

from sage.modular.modform_hecketriangle.space import ModularForms
MF = ModularForms(n=6, k=20, ep=1)
subspace = MF.subspace([(MF.Delta()*MF.E4()^2).as_ring_element(), MF.gen(0)])
subspace.gens()
subspace.gens()[0].parent() == subspace
rank()[source]

Return the rank of self.

EXAMPLES:

sage: from sage.modular.modform_hecketriangle.space import ModularForms
sage: MF = ModularForms(n=6, k=20, ep=1)
sage: subspace = MF.subspace([(MF.Delta()*MF.E4()^2).as_ring_element(), MF.gen(0)])
sage: subspace.rank()
2
sage: subspace.rank() == subspace.dimension()
True

>>> from sage.all import *
>>> from sage.modular.modform_hecketriangle.space import ModularForms
>>> MF = ModularForms(n=Integer(6), k=Integer(20), ep=Integer(1))
>>> subspace = MF.subspace([(MF.Delta()*MF.E4()**Integer(2)).as_ring_element(), MF.gen(Integer(0))])
>>> subspace.rank()
2
>>> subspace.rank() == subspace.dimension()
True

from sage.modular.modform_hecketriangle.space import ModularForms
MF = ModularForms(n=6, k=20, ep=1)
subspace = MF.subspace([(MF.Delta()*MF.E4()^2).as_ring_element(), MF.gen(0)])
subspace.rank()
subspace.rank() == subspace.dimension()
sage.modular.modform_hecketriangle.subspace.canonical_parameters(ambient_space, basis, check=True)[source]

Return a canonical version of the parameters. In particular the list/tuple basis is replaced by a tuple of linearly independent elements in the ambient space.

If check=False (default: True) then basis is assumed to already be a basis.

EXAMPLES:

sage: from sage.modular.modform_hecketriangle.subspace import canonical_parameters
sage: from sage.modular.modform_hecketriangle.space import ModularForms
sage: MF = ModularForms(n=6, k=12, ep=1)
sage: canonical_parameters(MF, [MF.Delta().as_ring_element(), MF.gen(0), 2*MF.gen(0)])
(ModularForms(n=6, k=12, ep=1) over Integer Ring,
 (q + 30*q^2 + 333*q^3 + 1444*q^4 + O(q^5),
  1 + 26208*q^3 + 530712*q^4 + O(q^5)))

>>> from sage.all import *
>>> from sage.modular.modform_hecketriangle.subspace import canonical_parameters
>>> from sage.modular.modform_hecketriangle.space import ModularForms
>>> MF = ModularForms(n=Integer(6), k=Integer(12), ep=Integer(1))
>>> canonical_parameters(MF, [MF.Delta().as_ring_element(), MF.gen(Integer(0)), Integer(2)*MF.gen(Integer(0))])
(ModularForms(n=6, k=12, ep=1) over Integer Ring,
 (q + 30*q^2 + 333*q^3 + 1444*q^4 + O(q^5),
  1 + 26208*q^3 + 530712*q^4 + O(q^5)))

from sage.modular.modform_hecketriangle.subspace import canonical_parameters
from sage.modular.modform_hecketriangle.space import ModularForms
MF = ModularForms(n=6, k=12, ep=1)
canonical_parameters(MF, [MF.Delta().as_ring_element(), MF.gen(0), 2*MF.gen(0)])