Elements of modular forms spaces

Class hierarchy:

• ModularForm_abstract

  • Newform

    • ModularFormElement_elliptic_curve

  • ModularFormElement

    • EisensteinSeries

• GradedModularFormElement

AUTHORS:

• William Stein (2004-2008): first version

• David Ayotte (2021-06): GradedModularFormElement class

class sage.modular.modform.element.EisensteinSeries(parent, vector, t, chi, psi)

Bases: ModularFormElement

An Eisenstein series.

EXAMPLES:

Sage

sage: E = EisensteinForms(1,12)
sage: E.eisenstein_series()
[
691/65520 + q + 2049*q^2 + 177148*q^3 + 4196353*q^4 + 48828126*q^5 + O(q^6)
]
sage: E = EisensteinForms(11,2)
sage: E.eisenstein_series()
[
5/12 + q + 3*q^2 + 4*q^3 + 7*q^4 + 6*q^5 + O(q^6)
]
sage: E = EisensteinForms(Gamma1(7),2)
sage: E.set_precision(4)
sage: E.eisenstein_series()
[
1/4 + q + 3*q^2 + 4*q^3 + O(q^4),
1/7*zeta6 - 3/7 + q + (-2*zeta6 + 1)*q^2 + (3*zeta6 - 2)*q^3 + O(q^4),
q + (-zeta6 + 2)*q^2 + (zeta6 + 2)*q^3 + O(q^4),
-1/7*zeta6 - 2/7 + q + (2*zeta6 - 1)*q^2 + (-3*zeta6 + 1)*q^3 + O(q^4),
q + (zeta6 + 1)*q^2 + (-zeta6 + 3)*q^3 + O(q^4)
]

Python

>>> from sage.all import *
>>> E = EisensteinForms(Integer(1),Integer(12))
>>> E.eisenstein_series()
[
691/65520 + q + 2049*q^2 + 177148*q^3 + 4196353*q^4 + 48828126*q^5 + O(q^6)
]
>>> E = EisensteinForms(Integer(11),Integer(2))
>>> E.eisenstein_series()
[
5/12 + q + 3*q^2 + 4*q^3 + 7*q^4 + 6*q^5 + O(q^6)
]
>>> E = EisensteinForms(Gamma1(Integer(7)),Integer(2))
>>> E.set_precision(Integer(4))
>>> E.eisenstein_series()
[
1/4 + q + 3*q^2 + 4*q^3 + O(q^4),
1/7*zeta6 - 3/7 + q + (-2*zeta6 + 1)*q^2 + (3*zeta6 - 2)*q^3 + O(q^4),
q + (-zeta6 + 2)*q^2 + (zeta6 + 2)*q^3 + O(q^4),
-1/7*zeta6 - 2/7 + q + (2*zeta6 - 1)*q^2 + (-3*zeta6 + 1)*q^3 + O(q^4),
q + (zeta6 + 1)*q^2 + (-zeta6 + 3)*q^3 + O(q^4)
]

Sage Live

E = EisensteinForms(1,12)
E.eisenstein_series()
E = EisensteinForms(11,2)
E.eisenstein_series()
E = EisensteinForms(Gamma1(7),2)
E.set_precision(4)
E.eisenstein_series()

L()

Return the conductor of self.chi().

EXAMPLES:

Sage

sage: EisensteinForms(DirichletGroup(17).0,99).eisenstein_series()[1].L()
17

Python

>>> from sage.all import *
>>> EisensteinForms(DirichletGroup(Integer(17)).gen(0),Integer(99)).eisenstein_series()[Integer(1)].L()
17

Sage Live

EisensteinForms(DirichletGroup(17).0,99).eisenstein_series()[1].L()

M()

Return the conductor of self.psi().

EXAMPLES:

Sage

sage: EisensteinForms(DirichletGroup(17).0,99).eisenstein_series()[1].M()
1

Python

>>> from sage.all import *
>>> EisensteinForms(DirichletGroup(Integer(17)).gen(0),Integer(99)).eisenstein_series()[Integer(1)].M()
1

Sage Live

EisensteinForms(DirichletGroup(17).0,99).eisenstein_series()[1].M()

character()

Return the character associated to self.

EXAMPLES:

Sage

sage: EisensteinForms(DirichletGroup(17).0,99).eisenstein_series()[1].character()
Dirichlet character modulo 17 of conductor 17 mapping 3 |--> zeta16

sage: chi = DirichletGroup(7)[4]
sage: E = EisensteinForms(chi).eisenstein_series() ; E
[
-1/7*zeta6 - 2/7 + q + (2*zeta6 - 1)*q^2 + (-3*zeta6 + 1)*q^3 + (-2*zeta6 - 1)*q^4 + (5*zeta6 - 4)*q^5 + O(q^6),
q + (zeta6 + 1)*q^2 + (-zeta6 + 3)*q^3 + (zeta6 + 2)*q^4 + (zeta6 + 4)*q^5 + O(q^6)
]
sage: E[0].character() == chi
True
sage: E[1].character() == chi
True

Python

>>> from sage.all import *
>>> EisensteinForms(DirichletGroup(Integer(17)).gen(0),Integer(99)).eisenstein_series()[Integer(1)].character()
Dirichlet character modulo 17 of conductor 17 mapping 3 |--> zeta16

>>> chi = DirichletGroup(Integer(7))[Integer(4)]
>>> E = EisensteinForms(chi).eisenstein_series() ; E
[
-1/7*zeta6 - 2/7 + q + (2*zeta6 - 1)*q^2 + (-3*zeta6 + 1)*q^3 + (-2*zeta6 - 1)*q^4 + (5*zeta6 - 4)*q^5 + O(q^6),
q + (zeta6 + 1)*q^2 + (-zeta6 + 3)*q^3 + (zeta6 + 2)*q^4 + (zeta6 + 4)*q^5 + O(q^6)
]
>>> E[Integer(0)].character() == chi
True
>>> E[Integer(1)].character() == chi
True

Sage Live

EisensteinForms(DirichletGroup(17).0,99).eisenstein_series()[1].character()
chi = DirichletGroup(7)[4]
E = EisensteinForms(chi).eisenstein_series() ; E
E[0].character() == chi
E[1].character() == chi

chi()

Return the parameter chi associated to self.

EXAMPLES:

Sage

sage: EisensteinForms(DirichletGroup(17).0,99).eisenstein_series()[1].chi()
Dirichlet character modulo 17 of conductor 17 mapping 3 |--> zeta16

Python

>>> from sage.all import *
>>> EisensteinForms(DirichletGroup(Integer(17)).gen(0),Integer(99)).eisenstein_series()[Integer(1)].chi()
Dirichlet character modulo 17 of conductor 17 mapping 3 |--> zeta16

Sage Live

EisensteinForms(DirichletGroup(17).0,99).eisenstein_series()[1].chi()

new_level()

Return level at which self is new.

EXAMPLES:

Sage

sage: EisensteinForms(DirichletGroup(17).0,99).eisenstein_series()[1].level()
17
sage: EisensteinForms(DirichletGroup(17).0,99).eisenstein_series()[1].new_level()
17
sage: [ [x.level(), x.new_level()] for x in EisensteinForms(DirichletGroup(60).0^2,2).eisenstein_series() ]
[[60, 2], [60, 3], [60, 2], [60, 5], [60, 2], [60, 2], [60, 2], [60, 3], [60, 2], [60, 2], [60, 2]]

Python

>>> from sage.all import *
>>> EisensteinForms(DirichletGroup(Integer(17)).gen(0),Integer(99)).eisenstein_series()[Integer(1)].level()
17
>>> EisensteinForms(DirichletGroup(Integer(17)).gen(0),Integer(99)).eisenstein_series()[Integer(1)].new_level()
17
>>> [ [x.level(), x.new_level()] for x in EisensteinForms(DirichletGroup(Integer(60)).gen(0)**Integer(2),Integer(2)).eisenstein_series() ]
[[60, 2], [60, 3], [60, 2], [60, 5], [60, 2], [60, 2], [60, 2], [60, 3], [60, 2], [60, 2], [60, 2]]

Sage Live

EisensteinForms(DirichletGroup(17).0,99).eisenstein_series()[1].level()
EisensteinForms(DirichletGroup(17).0,99).eisenstein_series()[1].new_level()
[ [x.level(), x.new_level()] for x in EisensteinForms(DirichletGroup(60).0^2,2).eisenstein_series() ]

parameters()

Return chi, psi, and t, which are the defining parameters of self.

EXAMPLES:

Sage

sage: EisensteinForms(DirichletGroup(17).0,99).eisenstein_series()[1].parameters()
(Dirichlet character modulo 17 of conductor 17 mapping 3 |--> zeta16, Dirichlet character modulo 17 of conductor 1 mapping 3 |--> 1, 1)

Python

>>> from sage.all import *
>>> EisensteinForms(DirichletGroup(Integer(17)).gen(0),Integer(99)).eisenstein_series()[Integer(1)].parameters()
(Dirichlet character modulo 17 of conductor 17 mapping 3 |--> zeta16, Dirichlet character modulo 17 of conductor 1 mapping 3 |--> 1, 1)

Sage Live

EisensteinForms(DirichletGroup(17).0,99).eisenstein_series()[1].parameters()

psi()

Return the parameter psi associated to self.

EXAMPLES:

Sage

sage: EisensteinForms(DirichletGroup(17).0,99).eisenstein_series()[1].psi()
 Dirichlet character modulo 17 of conductor 1 mapping 3 |--> 1

Python

>>> from sage.all import *
>>> EisensteinForms(DirichletGroup(Integer(17)).gen(0),Integer(99)).eisenstein_series()[Integer(1)].psi()
 Dirichlet character modulo 17 of conductor 1 mapping 3 |--> 1

Sage Live

EisensteinForms(DirichletGroup(17).0,99).eisenstein_series()[1].psi()

t()

Return the parameter t associated to self.

EXAMPLES:

Sage

sage: EisensteinForms(DirichletGroup(17).0,99).eisenstein_series()[1].t()
1

Python

>>> from sage.all import *
>>> EisensteinForms(DirichletGroup(Integer(17)).gen(0),Integer(99)).eisenstein_series()[Integer(1)].t()
1

Sage Live

EisensteinForms(DirichletGroup(17).0,99).eisenstein_series()[1].t()

class sage.modular.modform.element.GradedModularFormElement(parent, forms_datum)

Bases: ModuleElement

The element class for ModularFormsRing. A GradedModularFormElement is basically a formal sum of modular forms of different weight: \(f_1 + f_2 + ... + f_n\). Note that a GradedModularFormElement is not necessarily a modular form (as it can have mixed weight components).

A GradedModularFormElement should not be constructed directly via this class. Instead, one should use the element constructor of the parent class (ModularFormsRing).

EXAMPLES:

Sage

sage: M = ModularFormsRing(1)
sage: D = CuspForms(1, 12).0
sage: M(D).parent()
Ring of Modular Forms for Modular Group SL(2,Z) over Rational Field

Python

>>> from sage.all import *
>>> M = ModularFormsRing(Integer(1))
>>> D = CuspForms(Integer(1), Integer(12)).gen(0)
>>> M(D).parent()
Ring of Modular Forms for Modular Group SL(2,Z) over Rational Field

Sage Live

M = ModularFormsRing(1)
D = CuspForms(1, 12).0
M(D).parent()

A graded modular form can be initiated via a dictionary or a list:

Sage

sage: E4 = ModularForms(1, 4).0
sage: M({4:E4, 12:D})  # dictionary
1 + 241*q + 2136*q^2 + 6972*q^3 + 16048*q^4 + 35070*q^5 + O(q^6)
sage: M([E4, D])  # list
1 + 241*q + 2136*q^2 + 6972*q^3 + 16048*q^4 + 35070*q^5 + O(q^6)

Python

>>> from sage.all import *
>>> E4 = ModularForms(Integer(1), Integer(4)).gen(0)
>>> M({Integer(4):E4, Integer(12):D})  # dictionary
1 + 241*q + 2136*q^2 + 6972*q^3 + 16048*q^4 + 35070*q^5 + O(q^6)
>>> M([E4, D])  # list
1 + 241*q + 2136*q^2 + 6972*q^3 + 16048*q^4 + 35070*q^5 + O(q^6)

Sage Live

E4 = ModularForms(1, 4).0
M({4:E4, 12:D})  # dictionary
M([E4, D])  # list

Also, when adding two modular forms of different weights, a graded modular form element will be created:

Sage

sage: (E4 + D).parent()
Ring of Modular Forms for Modular Group SL(2,Z) over Rational Field
sage: M([E4, D]) == E4 + D
True

Python

>>> from sage.all import *
>>> (E4 + D).parent()
Ring of Modular Forms for Modular Group SL(2,Z) over Rational Field
>>> M([E4, D]) == E4 + D
True

Sage Live

(E4 + D).parent()
M([E4, D]) == E4 + D

Graded modular forms elements for congruence subgroups are also supported:

Sage

sage: M = ModularFormsRing(Gamma0(3))
sage: f = ModularForms(Gamma0(3), 4).0
sage: g = ModularForms(Gamma0(3), 2).0
sage: M([f, g])
2 + 12*q + 36*q^2 + 252*q^3 + 84*q^4 + 72*q^5 + O(q^6)
sage: M({4:f, 2:g})
2 + 12*q + 36*q^2 + 252*q^3 + 84*q^4 + 72*q^5 + O(q^6)

Python

>>> from sage.all import *
>>> M = ModularFormsRing(Gamma0(Integer(3)))
>>> f = ModularForms(Gamma0(Integer(3)), Integer(4)).gen(0)
>>> g = ModularForms(Gamma0(Integer(3)), Integer(2)).gen(0)
>>> M([f, g])
2 + 12*q + 36*q^2 + 252*q^3 + 84*q^4 + 72*q^5 + O(q^6)
>>> M({Integer(4):f, Integer(2):g})
2 + 12*q + 36*q^2 + 252*q^3 + 84*q^4 + 72*q^5 + O(q^6)

Sage Live

M = ModularFormsRing(Gamma0(3))
f = ModularForms(Gamma0(3), 4).0
g = ModularForms(Gamma0(3), 2).0
M([f, g])
M({4:f, 2:g})

derivative(name='E2')

Return the derivative \(q \frac{d}{dq}\) of the given graded form.

Note that this method returns an element of a new parent, that is a quasimodular form. If the form is not homogeneous, then this method sums the derivative of each homogeneous component.

INPUT:

• name– string (default: 'E2'); the name of the weight 2 Eisenstein series generating the graded algebra of quasimodular forms over the ring of modular forms

OUTPUT: a sage.modular.quasimodform.element.QuasiModularFormsElement

EXAMPLES:

Sage

sage: M = ModularFormsRing(1)
sage: E4 = M.0; E6 = M.1
sage: dE4 = E4.derivative(); dE4
240*q + 4320*q^2 + 20160*q^3 + 70080*q^4 + 151200*q^5 + O(q^6)
sage: dE4.parent()
Ring of Quasimodular Forms for Modular Group SL(2,Z) over Rational Field
sage: dE4.is_modular_form()
False

Python

>>> from sage.all import *
>>> M = ModularFormsRing(Integer(1))
>>> E4 = M.gen(0); E6 = M.gen(1)
>>> dE4 = E4.derivative(); dE4
240*q + 4320*q^2 + 20160*q^3 + 70080*q^4 + 151200*q^5 + O(q^6)
>>> dE4.parent()
Ring of Quasimodular Forms for Modular Group SL(2,Z) over Rational Field
>>> dE4.is_modular_form()
False

Sage Live

M = ModularFormsRing(1)
E4 = M.0; E6 = M.1
dE4 = E4.derivative(); dE4
dE4.parent()
dE4.is_modular_form()

group()

Return the group for which self is a modular form.

EXAMPLES:

Sage

sage: M = ModularFormsRing(1)
sage: E4 = M.0
sage: E4.group()
Modular Group SL(2,Z)
sage: M5 = ModularFormsRing(Gamma1(5))
sage: f = M5(ModularForms(Gamma1(5)).0);
sage: f.group()
Congruence Subgroup Gamma1(5)

Python

>>> from sage.all import *
>>> M = ModularFormsRing(Integer(1))
>>> E4 = M.gen(0)
>>> E4.group()
Modular Group SL(2,Z)
>>> M5 = ModularFormsRing(Gamma1(Integer(5)))
>>> f = M5(ModularForms(Gamma1(Integer(5))).gen(0));
>>> f.group()
Congruence Subgroup Gamma1(5)

Sage Live

M = ModularFormsRing(1)
E4 = M.0
E4.group()
M5 = ModularFormsRing(Gamma1(5))
f = M5(ModularForms(Gamma1(5)).0);
f.group()

homogeneous_component(weight)

Return the homogeneous component of the given graded modular form.

INPUT:

• weight – integer corresponding to the weight of the homogeneous component of the given graded modular form

EXAMPLES:

Sage

sage: M = ModularFormsRing(1)
sage: f4 = ModularForms(1, 4).0; f6 = ModularForms(1, 6).0; f8 = ModularForms(1, 8).0
sage: F = M(f4) + M(f6) + M(f8)
sage: F[4] # indirect doctest
1 + 240*q + 2160*q^2 + 6720*q^3 + 17520*q^4 + 30240*q^5 + O(q^6)
sage: F[6] # indirect doctest
1 - 504*q - 16632*q^2 - 122976*q^3 - 532728*q^4 - 1575504*q^5 + O(q^6)
sage: F[8] # indirect doctest
1 + 480*q + 61920*q^2 + 1050240*q^3 + 7926240*q^4 + 37500480*q^5 + O(q^6)
sage: F[10] # indirect doctest
0
sage: F.homogeneous_component(4)
1 + 240*q + 2160*q^2 + 6720*q^3 + 17520*q^4 + 30240*q^5 + O(q^6)

Python

>>> from sage.all import *
>>> M = ModularFormsRing(Integer(1))
>>> f4 = ModularForms(Integer(1), Integer(4)).gen(0); f6 = ModularForms(Integer(1), Integer(6)).gen(0); f8 = ModularForms(Integer(1), Integer(8)).gen(0)
>>> F = M(f4) + M(f6) + M(f8)
>>> F[Integer(4)] # indirect doctest
1 + 240*q + 2160*q^2 + 6720*q^3 + 17520*q^4 + 30240*q^5 + O(q^6)
>>> F[Integer(6)] # indirect doctest
1 - 504*q - 16632*q^2 - 122976*q^3 - 532728*q^4 - 1575504*q^5 + O(q^6)
>>> F[Integer(8)] # indirect doctest
1 + 480*q + 61920*q^2 + 1050240*q^3 + 7926240*q^4 + 37500480*q^5 + O(q^6)
>>> F[Integer(10)] # indirect doctest
0
>>> F.homogeneous_component(Integer(4))
1 + 240*q + 2160*q^2 + 6720*q^3 + 17520*q^4 + 30240*q^5 + O(q^6)

Sage Live

M = ModularFormsRing(1)
f4 = ModularForms(1, 4).0; f6 = ModularForms(1, 6).0; f8 = ModularForms(1, 8).0
F = M(f4) + M(f6) + M(f8)
F[4] # indirect doctest
F[6] # indirect doctest
F[8] # indirect doctest
F[10] # indirect doctest
F.homogeneous_component(4)

is_homogeneous()

Return True if the graded modular form is homogeneous, i.e. if it is a modular forms of a certain weight.

An alias of this method is is_modular_form

EXAMPLES:

Sage

sage: M = ModularFormsRing(1)
sage: E4 = M.0; E6 = M.1;
sage: E4.is_homogeneous()
True
sage: F = E4 + E6 # Not a modular form
sage: F.is_homogeneous()
False

Python

>>> from sage.all import *
>>> M = ModularFormsRing(Integer(1))
>>> E4 = M.gen(0); E6 = M.gen(1);
>>> E4.is_homogeneous()
True
>>> F = E4 + E6 # Not a modular form
>>> F.is_homogeneous()
False

Sage Live

M = ModularFormsRing(1)
E4 = M.0; E6 = M.1;
E4.is_homogeneous()
F = E4 + E6 # Not a modular form
F.is_homogeneous()

is_modular_form()

Return True if the graded modular form is homogeneous, i.e. if it is a modular forms of a certain weight.

An alias of this method is is_modular_form

EXAMPLES:

Sage

sage: M = ModularFormsRing(1)
sage: E4 = M.0; E6 = M.1;
sage: E4.is_homogeneous()
True
sage: F = E4 + E6 # Not a modular form
sage: F.is_homogeneous()
False

Python

>>> from sage.all import *
>>> M = ModularFormsRing(Integer(1))
>>> E4 = M.gen(0); E6 = M.gen(1);
>>> E4.is_homogeneous()
True
>>> F = E4 + E6 # Not a modular form
>>> F.is_homogeneous()
False

Sage Live

M = ModularFormsRing(1)
E4 = M.0; E6 = M.1;
E4.is_homogeneous()
F = E4 + E6 # Not a modular form
F.is_homogeneous()

is_one()

Return “True” if the graded form is 1 and “False” otherwise.

EXAMPLES:

Sage

sage: M = ModularFormsRing(1)
sage: M(1).is_one()
True
sage: M(2).is_one()
False
sage: E6 = M.0
sage: E6.is_one()
False

Python

>>> from sage.all import *
>>> M = ModularFormsRing(Integer(1))
>>> M(Integer(1)).is_one()
True
>>> M(Integer(2)).is_one()
False
>>> E6 = M.gen(0)
>>> E6.is_one()
False

Sage Live

M = ModularFormsRing(1)
M(1).is_one()
M(2).is_one()
E6 = M.0
E6.is_one()

is_zero()

Return “True” if the graded form is 0 and “False” otherwise.

EXAMPLES:

Sage

sage: M = ModularFormsRing(1)
sage: M(0).is_zero()
True
sage: M(1/2).is_zero()
False
sage: E6 = M.1
sage: M(E6).is_zero()
False

Python

>>> from sage.all import *
>>> M = ModularFormsRing(Integer(1))
>>> M(Integer(0)).is_zero()
True
>>> M(Integer(1)/Integer(2)).is_zero()
False
>>> E6 = M.gen(1)
>>> M(E6).is_zero()
False

Sage Live

M = ModularFormsRing(1)
M(0).is_zero()
M(1/2).is_zero()
E6 = M.1
M(E6).is_zero()

q_expansion(prec=None)

Return the \(q\)-expansion of the graded modular form up to precision prec (default: 6).

An alias of this method is qexp.

EXAMPLES:

Sage

sage: M = ModularFormsRing(1)
sage: zer = M(0); zer.q_expansion()
0
sage: M(5/7).q_expansion()
5/7
sage: E4 = M.0; E4
1 + 240*q + 2160*q^2 + 6720*q^3 + 17520*q^4 + 30240*q^5 + O(q^6)
sage: E6 = M.1; E6
1 - 504*q - 16632*q^2 - 122976*q^3 - 532728*q^4 - 1575504*q^5 + O(q^6)
sage: F = E4 + E6; F
2 - 264*q - 14472*q^2 - 116256*q^3 - 515208*q^4 - 1545264*q^5 + O(q^6)
sage: F.q_expansion()
2 - 264*q - 14472*q^2 - 116256*q^3 - 515208*q^4 - 1545264*q^5 + O(q^6)
sage: F.q_expansion(10)
2 - 264*q - 14472*q^2 - 116256*q^3 - 515208*q^4 - 1545264*q^5 - 3997728*q^6 - 8388672*q^7 - 16907400*q^8 - 29701992*q^9 + O(q^10)

Python

>>> from sage.all import *
>>> M = ModularFormsRing(Integer(1))
>>> zer = M(Integer(0)); zer.q_expansion()
0
>>> M(Integer(5)/Integer(7)).q_expansion()
5/7
>>> E4 = M.gen(0); E4
1 + 240*q + 2160*q^2 + 6720*q^3 + 17520*q^4 + 30240*q^5 + O(q^6)
>>> E6 = M.gen(1); E6
1 - 504*q - 16632*q^2 - 122976*q^3 - 532728*q^4 - 1575504*q^5 + O(q^6)
>>> F = E4 + E6; F
2 - 264*q - 14472*q^2 - 116256*q^3 - 515208*q^4 - 1545264*q^5 + O(q^6)
>>> F.q_expansion()
2 - 264*q - 14472*q^2 - 116256*q^3 - 515208*q^4 - 1545264*q^5 + O(q^6)
>>> F.q_expansion(Integer(10))
2 - 264*q - 14472*q^2 - 116256*q^3 - 515208*q^4 - 1545264*q^5 - 3997728*q^6 - 8388672*q^7 - 16907400*q^8 - 29701992*q^9 + O(q^10)

Sage Live

M = ModularFormsRing(1)
zer = M(0); zer.q_expansion()
M(5/7).q_expansion()
E4 = M.0; E4
E6 = M.1; E6
F = E4 + E6; F
F.q_expansion()
F.q_expansion(10)

qexp(prec=None)

Return the \(q\)-expansion of the graded modular form up to precision prec (default: 6).

An alias of this method is qexp.

EXAMPLES:

Sage

sage: M = ModularFormsRing(1)
sage: zer = M(0); zer.q_expansion()
0
sage: M(5/7).q_expansion()
5/7
sage: E4 = M.0; E4
1 + 240*q + 2160*q^2 + 6720*q^3 + 17520*q^4 + 30240*q^5 + O(q^6)
sage: E6 = M.1; E6
1 - 504*q - 16632*q^2 - 122976*q^3 - 532728*q^4 - 1575504*q^5 + O(q^6)
sage: F = E4 + E6; F
2 - 264*q - 14472*q^2 - 116256*q^3 - 515208*q^4 - 1545264*q^5 + O(q^6)
sage: F.q_expansion()
2 - 264*q - 14472*q^2 - 116256*q^3 - 515208*q^4 - 1545264*q^5 + O(q^6)
sage: F.q_expansion(10)
2 - 264*q - 14472*q^2 - 116256*q^3 - 515208*q^4 - 1545264*q^5 - 3997728*q^6 - 8388672*q^7 - 16907400*q^8 - 29701992*q^9 + O(q^10)

Python

>>> from sage.all import *
>>> M = ModularFormsRing(Integer(1))
>>> zer = M(Integer(0)); zer.q_expansion()
0
>>> M(Integer(5)/Integer(7)).q_expansion()
5/7
>>> E4 = M.gen(0); E4
1 + 240*q + 2160*q^2 + 6720*q^3 + 17520*q^4 + 30240*q^5 + O(q^6)
>>> E6 = M.gen(1); E6
1 - 504*q - 16632*q^2 - 122976*q^3 - 532728*q^4 - 1575504*q^5 + O(q^6)
>>> F = E4 + E6; F
2 - 264*q - 14472*q^2 - 116256*q^3 - 515208*q^4 - 1545264*q^5 + O(q^6)
>>> F.q_expansion()
2 - 264*q - 14472*q^2 - 116256*q^3 - 515208*q^4 - 1545264*q^5 + O(q^6)
>>> F.q_expansion(Integer(10))
2 - 264*q - 14472*q^2 - 116256*q^3 - 515208*q^4 - 1545264*q^5 - 3997728*q^6 - 8388672*q^7 - 16907400*q^8 - 29701992*q^9 + O(q^10)

Sage Live

M = ModularFormsRing(1)
zer = M(0); zer.q_expansion()
M(5/7).q_expansion()
E4 = M.0; E4
E6 = M.1; E6
F = E4 + E6; F
F.q_expansion()
F.q_expansion(10)

serre_derivative()

Return the Serre derivative of the given graded modular form.

If self is a modular form of weight \(k\), then the returned modular form will be of weight \(k + 2\). If the form is not homogeneous, then this method sums the Serre derivative of each homogeneous component.

EXAMPLES:

Sage

sage: M = ModularFormsRing(1)
sage: E4 = M.0
sage: E6 = M.1
sage: DE4 = E4.serre_derivative(); DE4
-1/3 + 168*q + 5544*q^2 + 40992*q^3 + 177576*q^4 + 525168*q^5 + O(q^6)
sage: DE4 == (-1/3) * E6
True
sage: DE6 = E6.serre_derivative(); DE6
-1/2 - 240*q - 30960*q^2 - 525120*q^3 - 3963120*q^4 - 18750240*q^5 + O(q^6)
sage: DE6 == (-1/2) * E4^2
True
sage: f = E4 + E6
sage: Df = f.serre_derivative(); Df
-5/6 - 72*q - 25416*q^2 - 484128*q^3 - 3785544*q^4 - 18225072*q^5 + O(q^6)
sage: Df == (-1/3) * E6 + (-1/2) * E4^2
True
sage: M(1/2).serre_derivative()
0

Python

>>> from sage.all import *
>>> M = ModularFormsRing(Integer(1))
>>> E4 = M.gen(0)
>>> E6 = M.gen(1)
>>> DE4 = E4.serre_derivative(); DE4
-1/3 + 168*q + 5544*q^2 + 40992*q^3 + 177576*q^4 + 525168*q^5 + O(q^6)
>>> DE4 == (-Integer(1)/Integer(3)) * E6
True
>>> DE6 = E6.serre_derivative(); DE6
-1/2 - 240*q - 30960*q^2 - 525120*q^3 - 3963120*q^4 - 18750240*q^5 + O(q^6)
>>> DE6 == (-Integer(1)/Integer(2)) * E4**Integer(2)
True
>>> f = E4 + E6
>>> Df = f.serre_derivative(); Df
-5/6 - 72*q - 25416*q^2 - 484128*q^3 - 3785544*q^4 - 18225072*q^5 + O(q^6)
>>> Df == (-Integer(1)/Integer(3)) * E6 + (-Integer(1)/Integer(2)) * E4**Integer(2)
True
>>> M(Integer(1)/Integer(2)).serre_derivative()
0

Sage Live

M = ModularFormsRing(1)
E4 = M.0
E6 = M.1
DE4 = E4.serre_derivative(); DE4
DE4 == (-1/3) * E6
DE6 = E6.serre_derivative(); DE6
DE6 == (-1/2) * E4^2
f = E4 + E6
Df = f.serre_derivative(); Df
Df == (-1/3) * E6 + (-1/2) * E4^2
M(1/2).serre_derivative()

to_polynomial(names='x', gens=None)

Return a polynomial \(P(x_0,..., x_n)\) such that \(P(g_0,..., g_n)\) is equal to self where \(g_0, ..., g_n\) is a list of generators of the parent.

INPUT:

• names – list or tuple of names (strings), or a comma separated string; corresponds to the names of the variables

• gens – (default: None) a list of generator of the parent of self. If set to None, the list returned by gen_forms() is used instead

OUTPUT: a polynomial in the variables names

EXAMPLES:

Sage

sage: M = ModularFormsRing(1)
sage: (M.0 + M.1).to_polynomial()
x1 + x0
sage: (M.0^10 + M.0 * M.1).to_polynomial()
x0^10 + x0*x1

Python

>>> from sage.all import *
>>> M = ModularFormsRing(Integer(1))
>>> (M.gen(0) + M.gen(1)).to_polynomial()
x1 + x0
>>> (M.gen(0)**Integer(10) + M.gen(0) * M.gen(1)).to_polynomial()
x0^10 + x0*x1

Sage Live

M = ModularFormsRing(1)
(M.0 + M.1).to_polynomial()
(M.0^10 + M.0 * M.1).to_polynomial()

This method is not necessarily the inverse of from_polynomial() since there may be some relations between the generators of the modular forms ring:

Sage

sage: M = ModularFormsRing(Gamma0(6))
sage: P.<x0,x1,x2> = M.polynomial_ring()
sage: M.from_polynomial(x1^2).to_polynomial()
x0*x2 + 2*x1*x2 + 11*x2^2

Python

>>> from sage.all import *
>>> M = ModularFormsRing(Gamma0(Integer(6)))
>>> P = M.polynomial_ring(names=('x0', 'x1', 'x2',)); (x0, x1, x2,) = P._first_ngens(3)
>>> M.from_polynomial(x1**Integer(2)).to_polynomial()
x0*x2 + 2*x1*x2 + 11*x2^2

Sage Live

M = ModularFormsRing(Gamma0(6))
P.<x0,x1,x2> = M.polynomial_ring()
M.from_polynomial(x1^2).to_polynomial()

weight()

Return the weight of the given form if it is homogeneous (i.e. a modular form).

EXAMPLES:

Sage

sage: D = ModularForms(1,12).0; M = ModularFormsRing(1)
sage: M(D).weight()
12
sage: M.zero().weight()
0
sage: e4 = ModularForms(1,4).0
sage: (M(D)+e4).weight()
Traceback (most recent call last):
...
ValueError: the given graded form is not homogeneous (not a modular form)

Python

>>> from sage.all import *
>>> D = ModularForms(Integer(1),Integer(12)).gen(0); M = ModularFormsRing(Integer(1))
>>> M(D).weight()
12
>>> M.zero().weight()
0
>>> e4 = ModularForms(Integer(1),Integer(4)).gen(0)
>>> (M(D)+e4).weight()
Traceback (most recent call last):
...
ValueError: the given graded form is not homogeneous (not a modular form)

Sage Live

D = ModularForms(1,12).0; M = ModularFormsRing(1)
M(D).weight()
M.zero().weight()
e4 = ModularForms(1,4).0
(M(D)+e4).weight()

weights_list()

Return the list of the weights of all the homogeneous components of the given graded modular form.

EXAMPLES:

Sage

sage: M = ModularFormsRing(1)
sage: f4 = ModularForms(1, 4).0; f6 = ModularForms(1, 6).0; f8 = ModularForms(1, 8).0
sage: F4 = M(f4); F6 = M(f6); F8 = M(f8)
sage: F = F4 + F6 + F8
sage: F.weights_list()
[4, 6, 8]
sage: M(0).weights_list()
[0]

Python

>>> from sage.all import *
>>> M = ModularFormsRing(Integer(1))
>>> f4 = ModularForms(Integer(1), Integer(4)).gen(0); f6 = ModularForms(Integer(1), Integer(6)).gen(0); f8 = ModularForms(Integer(1), Integer(8)).gen(0)
>>> F4 = M(f4); F6 = M(f6); F8 = M(f8)
>>> F = F4 + F6 + F8
>>> F.weights_list()
[4, 6, 8]
>>> M(Integer(0)).weights_list()
[0]

Sage Live

M = ModularFormsRing(1)
f4 = ModularForms(1, 4).0; f6 = ModularForms(1, 6).0; f8 = ModularForms(1, 8).0
F4 = M(f4); F6 = M(f6); F8 = M(f8)
F = F4 + F6 + F8
F.weights_list()
M(0).weights_list()

class sage.modular.modform.element.ModularFormElement(parent, x, check=True)

Bases: ModularForm_abstract, HeckeModuleElement

An element of a space of modular forms.

INPUT:

• parent – ModularFormsSpace (an ambient space of modular forms)

• x – a vector on the basis for parent

• check – if check is True, check the types of the inputs

OUTPUT: ModularFormElement – a modular form

EXAMPLES:

Sage

sage: M = ModularForms(Gamma0(11),2)
sage: f = M.0
sage: f.parent()
Modular Forms space of dimension 2 for Congruence Subgroup Gamma0(11) of weight 2 over Rational Field

Python

>>> from sage.all import *
>>> M = ModularForms(Gamma0(Integer(11)),Integer(2))
>>> f = M.gen(0)
>>> f.parent()
Modular Forms space of dimension 2 for Congruence Subgroup Gamma0(11) of weight 2 over Rational Field

Sage Live

M = ModularForms(Gamma0(11),2)
f = M.0
f.parent()

atkin_lehner_eigenvalue(d=None, embedding=None)

Return the result of the Atkin-Lehner operator \(W_d\) on self.

INPUT:

• d – positive integer exactly dividing the level \(N\) of self, i.e. \(d\) divides \(N\) and is coprime to \(N/d\). (default: \(d = N\))

• embedding – ignored (but accepted for compatibility with Newform.atkin_lehner_eigenvalue())

OUTPUT:

The Atkin-Lehner eigenvalue of \(W_d\) on self. If self is not an eigenform for \(W_d\), a ValueError is raised.

See also

For the conventions used to define the operator \(W_d\), see sage.modular.hecke.module.HeckeModule_free_module.atkin_lehner_operator().

EXAMPLES:

Sage

sage: CuspForms(1, 30).0.atkin_lehner_eigenvalue()
1
sage: CuspForms(2, 8).0.atkin_lehner_eigenvalue()
Traceback (most recent call last):
...
NotImplementedError: don't know how to compute Atkin-Lehner matrix acting on this space (try using a newform constructor instead)

Python

>>> from sage.all import *
>>> CuspForms(Integer(1), Integer(30)).gen(0).atkin_lehner_eigenvalue()
1
>>> CuspForms(Integer(2), Integer(8)).gen(0).atkin_lehner_eigenvalue()
Traceback (most recent call last):
...
NotImplementedError: don't know how to compute Atkin-Lehner matrix acting on this space (try using a newform constructor instead)

Sage Live

CuspForms(1, 30).0.atkin_lehner_eigenvalue()
CuspForms(2, 8).0.atkin_lehner_eigenvalue()

twist(chi, level=None)

Return the twist of the modular form self by the Dirichlet character chi.

If self is a modular form \(f\) with character \(\epsilon\) and \(q\)-expansion

\[f(q) = \sum_{n=0}^\infty a_n q^n,\]

then the twist by \(\chi\) is a modular form \(f_\chi\) with character \(\epsilon\chi^2\) and \(q\)-expansion

\[f_\chi(q) = \sum_{n=0}^\infty \chi(n) a_n q^n.\]

INPUT:

• chi – a Dirichlet character

• level – (optional) the level \(N\) of the twisted form. By default, the algorithm chooses some not necessarily minimal value for \(N\) using [AL1978], Proposition 3.1, (See also [Kob1993], Proposition III.3.17, for a simpler but slightly weaker bound.)

OUTPUT:

The form \(f_\chi\) as an element of the space of modular forms for \(\Gamma_1(N)\) with character \(\epsilon\chi^2\).

EXAMPLES:

Sage

sage: f = CuspForms(11, 2).0
sage: f.parent()
Cuspidal subspace of dimension 1 of Modular Forms space of dimension 2 for Congruence Subgroup Gamma0(11) of weight 2 over Rational Field
sage: f.q_expansion(6)
q - 2*q^2 - q^3 + 2*q^4 + q^5 + O(q^6)
sage: eps = DirichletGroup(3).0
sage: eps.parent()
Group of Dirichlet characters modulo 3 with values in Cyclotomic Field of order 2 and degree 1
sage: f_eps = f.twist(eps)
sage: f_eps.parent()
Cuspidal subspace of dimension 9 of Modular Forms space of dimension 16 for Congruence Subgroup Gamma0(99) of weight 2 over Cyclotomic Field of order 2 and degree 1
sage: f_eps.q_expansion(6)
q + 2*q^2 + 2*q^4 - q^5 + O(q^6)

Python

>>> from sage.all import *
>>> f = CuspForms(Integer(11), Integer(2)).gen(0)
>>> f.parent()
Cuspidal subspace of dimension 1 of Modular Forms space of dimension 2 for Congruence Subgroup Gamma0(11) of weight 2 over Rational Field
>>> f.q_expansion(Integer(6))
q - 2*q^2 - q^3 + 2*q^4 + q^5 + O(q^6)
>>> eps = DirichletGroup(Integer(3)).gen(0)
>>> eps.parent()
Group of Dirichlet characters modulo 3 with values in Cyclotomic Field of order 2 and degree 1
>>> f_eps = f.twist(eps)
>>> f_eps.parent()
Cuspidal subspace of dimension 9 of Modular Forms space of dimension 16 for Congruence Subgroup Gamma0(99) of weight 2 over Cyclotomic Field of order 2 and degree 1
>>> f_eps.q_expansion(Integer(6))
q + 2*q^2 + 2*q^4 - q^5 + O(q^6)

Sage Live

f = CuspForms(11, 2).0
f.parent()
f.q_expansion(6)
eps = DirichletGroup(3).0
eps.parent()
f_eps = f.twist(eps)
f_eps.parent()
f_eps.q_expansion(6)

Modular forms without character are supported:

Sage

sage: M = ModularForms(Gamma1(5), 2)
sage: f = M.gen(0); f
1 + 60*q^3 - 120*q^4 + 240*q^5 + O(q^6)
sage: chi = DirichletGroup(2)[0]
sage: f.twist(chi)
60*q^3 + 240*q^5 + O(q^6)

Python

>>> from sage.all import *
>>> M = ModularForms(Gamma1(Integer(5)), Integer(2))
>>> f = M.gen(Integer(0)); f
1 + 60*q^3 - 120*q^4 + 240*q^5 + O(q^6)
>>> chi = DirichletGroup(Integer(2))[Integer(0)]
>>> f.twist(chi)
60*q^3 + 240*q^5 + O(q^6)

Sage Live

M = ModularForms(Gamma1(5), 2)
f = M.gen(0); f
chi = DirichletGroup(2)[0]
f.twist(chi)

The base field of the twisted form is extended if necessary:

Sage

sage: E4 = ModularForms(1, 4).gen(0)
sage: E4.parent()
Modular Forms space of dimension 1 for Modular Group SL(2,Z) of weight 4 over Rational Field
sage: chi = DirichletGroup(5)[1]
sage: chi.base_ring()
Cyclotomic Field of order 4 and degree 2
sage: E4_chi = E4.twist(chi)
sage: E4_chi.parent()
Modular Forms space of dimension 10, character [-1] and weight 4 over Cyclotomic Field of order 4 and degree 2

Python

>>> from sage.all import *
>>> E4 = ModularForms(Integer(1), Integer(4)).gen(Integer(0))
>>> E4.parent()
Modular Forms space of dimension 1 for Modular Group SL(2,Z) of weight 4 over Rational Field
>>> chi = DirichletGroup(Integer(5))[Integer(1)]
>>> chi.base_ring()
Cyclotomic Field of order 4 and degree 2
>>> E4_chi = E4.twist(chi)
>>> E4_chi.parent()
Modular Forms space of dimension 10, character [-1] and weight 4 over Cyclotomic Field of order 4 and degree 2

Sage Live

E4 = ModularForms(1, 4).gen(0)
E4.parent()
chi = DirichletGroup(5)[1]
chi.base_ring()
E4_chi = E4.twist(chi)
E4_chi.parent()

REFERENCES:

• [AL1978]

• [Kob1993]

AUTHORS:

• L. J. P. Kilford (2009-08-28)

• Peter Bruin (2015-03-30)

class sage.modular.modform.element.ModularFormElement_elliptic_curve(parent, E)

Bases: Newform

A modular form attached to an elliptic curve over \(\QQ\).

atkin_lehner_eigenvalue(d=None, embedding=None)

Return the result of the Atkin-Lehner operator \(W_d\) on self.

INPUT:

• d – positive integer exactly dividing the level \(N\) of self, i.e. \(d\) divides \(N\) and is coprime to \(N/d\). (Defaults to \(d = N\) if not given.)

• embedding – ignored (but accepted for compatibility with Newform.atkin_lehner_action())

OUTPUT:

The Atkin-Lehner eigenvalue of \(W_d\) on self. This is either \(1\) or \(-1\).

EXAMPLES:

Sage

sage: EllipticCurve('57a1').newform().atkin_lehner_eigenvalue()
1
sage: EllipticCurve('57b1').newform().atkin_lehner_eigenvalue()
-1
sage: EllipticCurve('57b1').newform().atkin_lehner_eigenvalue(19)
1

Python

>>> from sage.all import *
>>> EllipticCurve('57a1').newform().atkin_lehner_eigenvalue()
1
>>> EllipticCurve('57b1').newform().atkin_lehner_eigenvalue()
-1
>>> EllipticCurve('57b1').newform().atkin_lehner_eigenvalue(Integer(19))
1

Sage Live

EllipticCurve('57a1').newform().atkin_lehner_eigenvalue()
EllipticCurve('57b1').newform().atkin_lehner_eigenvalue()
EllipticCurve('57b1').newform().atkin_lehner_eigenvalue(19)

elliptic_curve()

Return elliptic curve associated to self.

EXAMPLES:

Sage

sage: E = EllipticCurve('11a')
sage: f = E.modular_form()
sage: f.elliptic_curve()
Elliptic Curve defined by y^2 + y = x^3 - x^2 - 10*x - 20 over Rational Field
sage: f.elliptic_curve() is E
True

Python

>>> from sage.all import *
>>> E = EllipticCurve('11a')
>>> f = E.modular_form()
>>> f.elliptic_curve()
Elliptic Curve defined by y^2 + y = x^3 - x^2 - 10*x - 20 over Rational Field
>>> f.elliptic_curve() is E
True

Sage Live

E = EllipticCurve('11a')
f = E.modular_form()
f.elliptic_curve()
f.elliptic_curve() is E

class sage.modular.modform.element.ModularForm_abstract

Bases: ModuleElement

Constructor for generic class of a modular form. This should never be called directly; instead one should instantiate one of the derived classes of this class.

atkin_lehner_eigenvalue(d=None, embedding=None)

Return the eigenvalue of the Atkin-Lehner operator \(W_d\) acting on self.

INPUT:

• d – positive integer exactly dividing the level \(N\) of self, i.e. \(d\) divides \(N\) and is coprime to \(N/d\) (default: \(d = N\))

• embedding – (optional) embedding of the base ring of self into another ring

OUTPUT:

The Atkin-Lehner eigenvalue of \(W_d\) on self. This is returned as an element of the codomain of embedding if specified, and in (a suitable extension of) the base field of self otherwise.

If self is not an eigenform for \(W_d\), a ValueError is raised.

See also

sage.modular.hecke.module.HeckeModule_free_module.atkin_lehner_operator() (especially for the conventions used to define the operator \(W_d\)).

EXAMPLES:

Sage

sage: CuspForms(1, 12).0.atkin_lehner_eigenvalue()
1
sage: CuspForms(2, 8).0.atkin_lehner_eigenvalue()
Traceback (most recent call last):
...
NotImplementedError: don't know how to compute Atkin-Lehner matrix acting on this space (try using a newform constructor instead)

Python

>>> from sage.all import *
>>> CuspForms(Integer(1), Integer(12)).gen(0).atkin_lehner_eigenvalue()
1
>>> CuspForms(Integer(2), Integer(8)).gen(0).atkin_lehner_eigenvalue()
Traceback (most recent call last):
...
NotImplementedError: don't know how to compute Atkin-Lehner matrix acting on this space (try using a newform constructor instead)

Sage Live

CuspForms(1, 12).0.atkin_lehner_eigenvalue()
CuspForms(2, 8).0.atkin_lehner_eigenvalue()

character(compute=True)

Return the character of self. If compute=False, then this will return None unless the form was explicitly created as an element of a space of forms with character, skipping the (potentially expensive) computation of the matrices of the diamond operators.

EXAMPLES:

Sage

sage: ModularForms(DirichletGroup(17).0^2,2).2.character()
Dirichlet character modulo 17 of conductor 17 mapping 3 |--> zeta8

sage: CuspForms(Gamma1(7), 3).gen(0).character()
Dirichlet character modulo 7 of conductor 7 mapping 3 |--> -1
sage: CuspForms(Gamma1(7), 3).gen(0).character(compute = False) is None
True
sage: M = CuspForms(Gamma1(7), 5).gen(0).character()
Traceback (most recent call last):
...
ValueError: Form is not an eigenvector for <3>

Python

>>> from sage.all import *
>>> ModularForms(DirichletGroup(Integer(17)).gen(0)**Integer(2),Integer(2)).gen(2).character()
Dirichlet character modulo 17 of conductor 17 mapping 3 |--> zeta8

>>> CuspForms(Gamma1(Integer(7)), Integer(3)).gen(Integer(0)).character()
Dirichlet character modulo 7 of conductor 7 mapping 3 |--> -1
>>> CuspForms(Gamma1(Integer(7)), Integer(3)).gen(Integer(0)).character(compute = False) is None
True
>>> M = CuspForms(Gamma1(Integer(7)), Integer(5)).gen(Integer(0)).character()
Traceback (most recent call last):
...
ValueError: Form is not an eigenvector for <3>

Sage Live

ModularForms(DirichletGroup(17).0^2,2).2.character()
CuspForms(Gamma1(7), 3).gen(0).character()
CuspForms(Gamma1(7), 3).gen(0).character(compute = False) is None
M = CuspForms(Gamma1(7), 5).gen(0).character()

cm_discriminant()

Return the discriminant of the CM field associated to this form. An error will be raised if the form isn’t of CM type.

EXAMPLES:

Sage

sage: Newforms(49, 2)[0].cm_discriminant()
-7
sage: CuspForms(1, 12).gen(0).cm_discriminant()
Traceback (most recent call last):
...
ValueError: Not a CM form

Python

>>> from sage.all import *
>>> Newforms(Integer(49), Integer(2))[Integer(0)].cm_discriminant()
-7
>>> CuspForms(Integer(1), Integer(12)).gen(Integer(0)).cm_discriminant()
Traceback (most recent call last):
...
ValueError: Not a CM form

Sage Live

Newforms(49, 2)[0].cm_discriminant()
CuspForms(1, 12).gen(0).cm_discriminant()

coefficient(n)

Return the \(n\)-th coefficient of the \(q\)-expansion of self.

INPUT:

• n – nonnegative integer

EXAMPLES:

Sage

sage: f = ModularForms(1, 12).0; f
q - 24*q^2 + 252*q^3 - 1472*q^4 + 4830*q^5 + O(q^6)
sage: f.coefficient(0)
0
sage: f.coefficient(1)
1
sage: f.coefficient(2)
-24
sage: f.coefficient(3)
252
sage: f.coefficient(4)
-1472

Python

>>> from sage.all import *
>>> f = ModularForms(Integer(1), Integer(12)).gen(0); f
q - 24*q^2 + 252*q^3 - 1472*q^4 + 4830*q^5 + O(q^6)
>>> f.coefficient(Integer(0))
0
>>> f.coefficient(Integer(1))
1
>>> f.coefficient(Integer(2))
-24
>>> f.coefficient(Integer(3))
252
>>> f.coefficient(Integer(4))
-1472

Sage Live

f = ModularForms(1, 12).0; f
f.coefficient(0)
f.coefficient(1)
f.coefficient(2)
f.coefficient(3)
f.coefficient(4)

coefficients(X)

The coefficients a_n of self, for integers n>=0 in the list X. If X is an Integer, return coefficients for indices from 1 to X.

This function caches the results of the compute function.

group()

Return the group for which self is a modular form.

EXAMPLES:

Sage

sage: ModularForms(Gamma1(11), 2).gen(0).group()
Congruence Subgroup Gamma1(11)

Python

>>> from sage.all import *
>>> ModularForms(Gamma1(Integer(11)), Integer(2)).gen(Integer(0)).group()
Congruence Subgroup Gamma1(11)

Sage Live

ModularForms(Gamma1(11), 2).gen(0).group()

has_cm()

Return whether the modular form self has complex multiplication.

OUTPUT: boolean

See also

• cm_discriminant() (to return the CM field)

• sage.schemes.elliptic_curves.ell_rational_field.has_cm()

EXAMPLES:

Sage

sage: G = DirichletGroup(21); eps = G.0 * G.1
sage: Newforms(eps, 2)[0].has_cm()
True

Python

>>> from sage.all import *
>>> G = DirichletGroup(Integer(21)); eps = G.gen(0) * G.gen(1)
>>> Newforms(eps, Integer(2))[Integer(0)].has_cm()
True

Sage Live

G = DirichletGroup(21); eps = G.0 * G.1
Newforms(eps, 2)[0].has_cm()

This example illustrates what happens when candidate_characters(self) is the empty list.

Sage

sage: M = ModularForms(Gamma0(1), 12)
sage: C = M.cuspidal_submodule()
sage: Delta = C.gens()[0]
sage: Delta.has_cm()
False

Python

>>> from sage.all import *
>>> M = ModularForms(Gamma0(Integer(1)), Integer(12))
>>> C = M.cuspidal_submodule()
>>> Delta = C.gens()[Integer(0)]
>>> Delta.has_cm()
False

Sage Live

M = ModularForms(Gamma0(1), 12)
C = M.cuspidal_submodule()
Delta = C.gens()[0]
Delta.has_cm()

We now compare the function has_cm between elliptic curves and their associated modular forms.

Sage

sage: E = EllipticCurve([-1, 0])
sage: f = E.modular_form()
sage: f.has_cm()
True
sage: E.has_cm() == f.has_cm()
True

Python

>>> from sage.all import *
>>> E = EllipticCurve([-Integer(1), Integer(0)])
>>> f = E.modular_form()
>>> f.has_cm()
True
>>> E.has_cm() == f.has_cm()
True

Sage Live

E = EllipticCurve([-1, 0])
f = E.modular_form()
f.has_cm()
E.has_cm() == f.has_cm()

Here is a non-cm example coming from elliptic curves.

Sage

sage: E = EllipticCurve('11a')
sage: f = E.modular_form()
sage: f.has_cm()
False
sage: E.has_cm() == f.has_cm()
True

Python

>>> from sage.all import *
>>> E = EllipticCurve('11a')
>>> f = E.modular_form()
>>> f.has_cm()
False
>>> E.has_cm() == f.has_cm()
True

Sage Live

E = EllipticCurve('11a')
f = E.modular_form()
f.has_cm()
E.has_cm() == f.has_cm()

is_homogeneous()

Return True.

For compatibility with elements of a graded modular forms ring.

An alias of this method is is_modular_form.

See also

sage.modular.modform.element.GradedModularFormElement.is_homogeneous()

EXAMPLES:

Sage

sage: ModularForms(1,12).0.is_homogeneous()
True

Python

>>> from sage.all import *
>>> ModularForms(Integer(1),Integer(12)).gen(0).is_homogeneous()
True

Sage Live

ModularForms(1,12).0.is_homogeneous()

is_modular_form()

Return True.

For compatibility with elements of a graded modular forms ring.

An alias of this method is is_modular_form.

See also

sage.modular.modform.element.GradedModularFormElement.is_homogeneous()

EXAMPLES:

Sage

sage: ModularForms(1,12).0.is_homogeneous()
True

Python

>>> from sage.all import *
>>> ModularForms(Integer(1),Integer(12)).gen(0).is_homogeneous()
True

Sage Live

ModularForms(1,12).0.is_homogeneous()

level()

Return the level of self.

EXAMPLES:

Sage

sage: ModularForms(25,4).0.level()
25

Python

>>> from sage.all import *
>>> ModularForms(Integer(25),Integer(4)).gen(0).level()
25

Sage Live

ModularForms(25,4).0.level()

lseries(embedding=0, prec=53, max_imaginary_part=0, max_asymp_coeffs=40)

Return the \(L\)-series of the weight k cusp form \(f\) on \(\Gamma_0(N)\).

This actually returns an interface to Tim Dokchitser’s program for computing with the \(L\)-series of the cusp form.

INPUT:

• embedding – either an embedding of the coefficient field of self into \(\CC\), or an integer \(i\) between 0 and D-1 where D is the degree of the coefficient field (meaning to pick the \(i\)-th embedding). (default: 0)

• prec – integer (default: 53); bits precision

• max_imaginary_part – real number (default: 0)

• max_asymp_coeffs – integer (default: 40)

For more information on the significance of the last three arguments, see dokchitser.

Note

If an explicit embedding is given, but this embedding is specified to smaller precision than prec, it will be automatically refined to precision prec.

OUTPUT:

The \(L\)-series of the cusp form, as a sage.lfunctions.dokchitser.Dokchitser object.

EXAMPLES:

Sage

sage: f = CuspForms(2,8).newforms()[0]
sage: L = f.lseries()
sage: L
L-series associated to the cusp form q - 8*q^2 + 12*q^3 + 64*q^4 - 210*q^5 + O(q^6)
sage: L(1)
0.0884317737041015
sage: L(0.5)
0.0296568512531983

Python

>>> from sage.all import *
>>> f = CuspForms(Integer(2),Integer(8)).newforms()[Integer(0)]
>>> L = f.lseries()
>>> L
L-series associated to the cusp form q - 8*q^2 + 12*q^3 + 64*q^4 - 210*q^5 + O(q^6)
>>> L(Integer(1))
0.0884317737041015
>>> L(RealNumber('0.5'))
0.0296568512531983

Sage Live

f = CuspForms(2,8).newforms()[0]
L = f.lseries()
L
L(1)
L(0.5)

As a consistency check, we verify that the functional equation holds:

Sage

sage: abs(L.check_functional_equation()) < 1.0e-20
True

Python

>>> from sage.all import *
>>> abs(L.check_functional_equation()) < RealNumber('1.0e-20')
True

Sage Live

abs(L.check_functional_equation()) < 1.0e-20

For non-rational newforms we can specify an embedding of the coefficient field:

Sage

sage: f = Newforms(43, names='a')[1]
sage: K = f.hecke_eigenvalue_field()
sage: phi1, phi2 = K.embeddings(CC)
sage: L = f.lseries(embedding=phi1)
sage: L
L-series associated to the cusp form q + a1*q^2 - a1*q^3 + (-a1 + 2)*q^5 + O(q^6), a1=-1.41421356237310
sage: L(1)
0.620539857407845
sage: L = f.lseries(embedding=1)
sage: L(1)
0.921328017272472

Python

>>> from sage.all import *
>>> f = Newforms(Integer(43), names='a')[Integer(1)]
>>> K = f.hecke_eigenvalue_field()
>>> phi1, phi2 = K.embeddings(CC)
>>> L = f.lseries(embedding=phi1)
>>> L
L-series associated to the cusp form q + a1*q^2 - a1*q^3 + (-a1 + 2)*q^5 + O(q^6), a1=-1.41421356237310
>>> L(Integer(1))
0.620539857407845
>>> L = f.lseries(embedding=Integer(1))
>>> L(Integer(1))
0.921328017272472

Sage Live

f = Newforms(43, names='a')[1]
K = f.hecke_eigenvalue_field()
phi1, phi2 = K.embeddings(CC)
L = f.lseries(embedding=phi1)
L
L(1)
L = f.lseries(embedding=1)
L(1)

An example with a non-real coefficient field (\(\QQ(\zeta_3)\) in this case):

Sage

sage: f = Newforms(Gamma1(13), 2, names='a')[0]
sage: f.lseries(embedding=0)(1)
0.298115272465799 - 0.0402203326076734*I
sage: f.lseries(embedding=1)(1)
0.298115272465799 + 0.0402203326076732*I

Python

>>> from sage.all import *
>>> f = Newforms(Gamma1(Integer(13)), Integer(2), names='a')[Integer(0)]
>>> f.lseries(embedding=Integer(0))(Integer(1))
0.298115272465799 - 0.0402203326076734*I
>>> f.lseries(embedding=Integer(1))(Integer(1))
0.298115272465799 + 0.0402203326076732*I

Sage Live

f = Newforms(Gamma1(13), 2, names='a')[0]
f.lseries(embedding=0)(1)
f.lseries(embedding=1)(1)

We compute with the \(L\)-series of the Eisenstein series \(E_4\):

Sage

sage: f = ModularForms(1,4).0
sage: L = f.lseries()
sage: L(1)
-0.0304484570583933
sage: L = eisenstein_series_lseries(4)
sage: L(1)
-0.0304484570583933

Python

>>> from sage.all import *
>>> f = ModularForms(Integer(1),Integer(4)).gen(0)
>>> L = f.lseries()
>>> L(Integer(1))
-0.0304484570583933
>>> L = eisenstein_series_lseries(Integer(4))
>>> L(Integer(1))
-0.0304484570583933

Sage Live

f = ModularForms(1,4).0
L = f.lseries()
L(1)
L = eisenstein_series_lseries(4)
L(1)

Consistency check with delta_lseries (which computes coefficients in pari):

Sage

sage: delta = CuspForms(1,12).0
sage: L = delta.lseries()
sage: L(1)
0.0374412812685155
sage: L = delta_lseries()
sage: L(1)
0.0374412812685155

Python

>>> from sage.all import *
>>> delta = CuspForms(Integer(1),Integer(12)).gen(0)
>>> L = delta.lseries()
>>> L(Integer(1))
0.0374412812685155
>>> L = delta_lseries()
>>> L(Integer(1))
0.0374412812685155

Sage Live

delta = CuspForms(1,12).0
L = delta.lseries()
L(1)
L = delta_lseries()
L(1)

We check that Issue #5262 is fixed:

Sage

sage: E = EllipticCurve('37b2')
sage: h = Newforms(37)[1]
sage: Lh = h.lseries()
sage: LE = E.lseries()
sage: Lh(1), LE(1)
(0.725681061936153, 0.725681061936153)
sage: CuspForms(1, 30).0.lseries().eps
-1.00000000000000

Python

>>> from sage.all import *
>>> E = EllipticCurve('37b2')
>>> h = Newforms(Integer(37))[Integer(1)]
>>> Lh = h.lseries()
>>> LE = E.lseries()
>>> Lh(Integer(1)), LE(Integer(1))
(0.725681061936153, 0.725681061936153)
>>> CuspForms(Integer(1), Integer(30)).gen(0).lseries().eps
-1.00000000000000

Sage Live

E = EllipticCurve('37b2')
h = Newforms(37)[1]
Lh = h.lseries()
LE = E.lseries()
Lh(1), LE(1)
CuspForms(1, 30).0.lseries().eps

We check that Issue #25369 is fixed:

Sage

sage: f5 = Newforms(Gamma1(4), 5, names='a')[0]; f5
q - 4*q^2 + 16*q^4 - 14*q^5 + O(q^6)
sage: L5 = f5.lseries()
sage: abs(L5.check_functional_equation()) < 1e-15
True
sage: abs(L5(4) - (gamma(1/4)^8/(3840*pi^2)).n()) < 1e-15
True

Python

>>> from sage.all import *
>>> f5 = Newforms(Gamma1(Integer(4)), Integer(5), names='a')[Integer(0)]; f5
q - 4*q^2 + 16*q^4 - 14*q^5 + O(q^6)
>>> L5 = f5.lseries()
>>> abs(L5.check_functional_equation()) < RealNumber('1e-15')
True
>>> abs(L5(Integer(4)) - (gamma(Integer(1)/Integer(4))**Integer(8)/(Integer(3840)*pi**Integer(2))).n()) < RealNumber('1e-15')
True

Sage Live

f5 = Newforms(Gamma1(4), 5, names='a')[0]; f5
L5 = f5.lseries()
abs(L5.check_functional_equation()) < 1e-15
abs(L5(4) - (gamma(1/4)^8/(3840*pi^2)).n()) < 1e-15

We can change the precision (in bits):

Sage

sage: f = Newforms(389, names='a')[0]
sage: L = f.lseries(prec=30)
sage: abs(L(1)) < 2^-30
True
sage: L = f.lseries(prec=53)
sage: abs(L(1)) < 2^-53
True
sage: L = f.lseries(prec=100)
sage: abs(L(1)) < 2^-100
True

sage: f = Newforms(27, names='a')[0]
sage: L = f.lseries()
sage: L(1)
0.588879583428483

Python

>>> from sage.all import *
>>> f = Newforms(Integer(389), names='a')[Integer(0)]
>>> L = f.lseries(prec=Integer(30))
>>> abs(L(Integer(1))) < Integer(2)**-Integer(30)
True
>>> L = f.lseries(prec=Integer(53))
>>> abs(L(Integer(1))) < Integer(2)**-Integer(53)
True
>>> L = f.lseries(prec=Integer(100))
>>> abs(L(Integer(1))) < Integer(2)**-Integer(100)
True

>>> f = Newforms(Integer(27), names='a')[Integer(0)]
>>> L = f.lseries()
>>> L(Integer(1))
0.588879583428483

Sage Live

f = Newforms(389, names='a')[0]
L = f.lseries(prec=30)
abs(L(1)) < 2^-30
L = f.lseries(prec=53)
abs(L(1)) < 2^-53
L = f.lseries(prec=100)
abs(L(1)) < 2^-100
f = Newforms(27, names='a')[0]
L = f.lseries()
L(1)

padded_list(n)

Return a list of length n whose entries are the first n coefficients of the \(q\)-expansion of self.

EXAMPLES:

Sage

sage: CuspForms(1,12).0.padded_list(20)
[0, 1, -24, 252, -1472, 4830, -6048, -16744, 84480, -113643,
 -115920, 534612, -370944, -577738, 401856, 1217160, 987136,
 -6905934, 2727432, 10661420]

Python

>>> from sage.all import *
>>> CuspForms(Integer(1),Integer(12)).gen(0).padded_list(Integer(20))
[0, 1, -24, 252, -1472, 4830, -6048, -16744, 84480, -113643,
 -115920, 534612, -370944, -577738, 401856, 1217160, 987136,
 -6905934, 2727432, 10661420]

Sage Live

CuspForms(1,12).0.padded_list(20)

period(M, prec=53)

Return the period of self with respect to \(M\).

INPUT:

• self – a cusp form \(f\) of weight 2 for \(Gamma_0(N)\)

• M – an element of \(\Gamma_0(N)\)

• prec – (default: 53) the working precision in bits. If \(f\) is a normalised eigenform, then the output is correct to approximately this number of bits.

OUTPUT:

A numerical approximation of the period \(P_f(M)\). This period is defined by the following integral over the complex upper half-plane, for any \(\alpha\) in \(\Bold{P}^1(\QQ)\):

\[P_f(M) = 2 \pi i \int_\alpha^{M(\alpha)} f(z) dz.\]

This is independent of the choice of \(\alpha\).

EXAMPLES:

Sage

sage: C = Newforms(11, 2)[0]
sage: m = C.group()(matrix([[-4, -3], [11, 8]]))
sage: C.period(m)
-0.634604652139776 - 1.45881661693850*I

sage: f = Newforms(15, 2)[0]
sage: g = Gamma0(15)(matrix([[-4, -3], [15, 11]]))
sage: f.period(g)  # abs tol 1e-15
2.17298044293747e-16 - 1.59624222213178*I

Python

>>> from sage.all import *
>>> C = Newforms(Integer(11), Integer(2))[Integer(0)]
>>> m = C.group()(matrix([[-Integer(4), -Integer(3)], [Integer(11), Integer(8)]]))
>>> C.period(m)
-0.634604652139776 - 1.45881661693850*I

>>> f = Newforms(Integer(15), Integer(2))[Integer(0)]
>>> g = Gamma0(Integer(15))(matrix([[-Integer(4), -Integer(3)], [Integer(15), Integer(11)]]))
>>> f.period(g)  # abs tol 1e-15
2.17298044293747e-16 - 1.59624222213178*I

Sage Live

C = Newforms(11, 2)[0]
m = C.group()(matrix([[-4, -3], [11, 8]]))
C.period(m)
f = Newforms(15, 2)[0]
g = Gamma0(15)(matrix([[-4, -3], [15, 11]]))
f.period(g)  # abs tol 1e-15

If \(E\) is an elliptic curve over \(\QQ\) and \(f\) is the newform associated to \(E\), then the periods of \(f\) are in the period lattice of \(E\) up to an integer multiple:

Sage

sage: E = EllipticCurve('11a3')
sage: f = E.newform()
sage: g = Gamma0(11)([3, 1, 11, 4])
sage: f.period(g)
0.634604652139777 + 1.45881661693850*I
sage: omega1, omega2 = E.period_lattice().basis()
sage: -2/5*omega1 + omega2
0.634604652139777 + 1.45881661693850*I

Python

>>> from sage.all import *
>>> E = EllipticCurve('11a3')
>>> f = E.newform()
>>> g = Gamma0(Integer(11))([Integer(3), Integer(1), Integer(11), Integer(4)])
>>> f.period(g)
0.634604652139777 + 1.45881661693850*I
>>> omega1, omega2 = E.period_lattice().basis()
>>> -Integer(2)/Integer(5)*omega1 + omega2
0.634604652139777 + 1.45881661693850*I

Sage Live

E = EllipticCurve('11a3')
f = E.newform()
g = Gamma0(11)([3, 1, 11, 4])
f.period(g)
omega1, omega2 = E.period_lattice().basis()
-2/5*omega1 + omega2

The integer multiple is 5 in this case, which is explained by the fact that there is a 5-isogeny between the elliptic curves \(J_0(5)\) and \(E\).

The elliptic curve \(E\) has a pair of modular symbols attached to it, which can be computed using the method sage.schemes.elliptic_curves.ell_rational_field.EllipticCurve_rational_field.modular_symbol(). These can be used to express the periods of \(f\) as exact linear combinations of the real and the imaginary period of \(E\):

Sage

sage: s = E.modular_symbol(sign=+1)
sage: t = E.modular_symbol(sign=-1, implementation='sage')
sage: s(3/11), t(3/11)
(1/10, 1/2)
sage: s(3/11)*omega1 + t(3/11)*2*omega2.imag()*I
0.634604652139777 + 1.45881661693850*I

Python

>>> from sage.all import *
>>> s = E.modular_symbol(sign=+Integer(1))
>>> t = E.modular_symbol(sign=-Integer(1), implementation='sage')
>>> s(Integer(3)/Integer(11)), t(Integer(3)/Integer(11))
(1/10, 1/2)
>>> s(Integer(3)/Integer(11))*omega1 + t(Integer(3)/Integer(11))*Integer(2)*omega2.imag()*I
0.634604652139777 + 1.45881661693850*I

Sage Live

s = E.modular_symbol(sign=+1)
t = E.modular_symbol(sign=-1, implementation='sage')
s(3/11), t(3/11)
s(3/11)*omega1 + t(3/11)*2*omega2.imag()*I

ALGORITHM:

We use the series expression from [Cre1997], Chapter II, Proposition 2.10.3. The algorithm sums the first \(T\) terms of this series, where \(T\) is chosen in such a way that the result would approximate \(P_f(M)\) with an absolute error of at most \(2^{-\text{prec}}\) if all computations were done exactly.

Since the actual precision is finite, the output is currently not guaranteed to be correct to prec bits of precision.

petersson_norm(embedding=0, prec=53)

Compute the Petersson scalar product of f with itself:

\[\langle f, f \rangle = \int_{\Gamma_0(N) \backslash \mathbb{H}} |f(x + iy)|^2 y^k\, \mathrm{d}x\, \mathrm{d}y.\]

Only implemented for N = 1 at present. It is assumed that \(f\) has real coefficients. The norm is computed as a special value of the symmetric square \(L\)-function, using the identity

\[\langle f, f \rangle = \frac{(k-1)! L(\mathrm{Sym}^2 f, k)}{2^{2k-1} \pi^{k+1}}\]

INPUT:

• embedding – embedding of the coefficient field into \(\RR\) or \(\CC\), or an integer \(i\) (interpreted as the \(i\)-th embedding) (default: 0)

• prec – integer (default: 53); precision in bits

EXAMPLES:

Sage

sage: CuspForms(1, 16).0.petersson_norm()
verbose -1 (...: dokchitser.py, __call__) Warning: Loss of 2 decimal digits due to cancellation
2.16906134759063e-6

Python

>>> from sage.all import *
>>> CuspForms(Integer(1), Integer(16)).gen(0).petersson_norm()
verbose -1 (...: dokchitser.py, __call__) Warning: Loss of 2 decimal digits due to cancellation
2.16906134759063e-6

Sage Live

CuspForms(1, 16).0.petersson_norm()

The Petersson norm depends on a choice of embedding:

Sage

sage: set_verbose(-2, "dokchitser.py") # disable precision-loss warnings
sage: F = Newforms(1, 24, names='a')[0]
sage: F.petersson_norm(embedding=0)
0.000107836545077234
sage: F.petersson_norm(embedding=1)
0.000128992800758160

Python

>>> from sage.all import *
>>> set_verbose(-Integer(2), "dokchitser.py") # disable precision-loss warnings
>>> F = Newforms(Integer(1), Integer(24), names='a')[Integer(0)]
>>> F.petersson_norm(embedding=Integer(0))
0.000107836545077234
>>> F.petersson_norm(embedding=Integer(1))
0.000128992800758160

Sage Live

set_verbose(-2, "dokchitser.py") # disable precision-loss warnings
F = Newforms(1, 24, names='a')[0]
F.petersson_norm(embedding=0)
F.petersson_norm(embedding=1)

prec()

Return the precision to which self.q_expansion() is currently known. Note that this may be 0.

EXAMPLES:

Sage

sage: M = ModularForms(2,14)
sage: f = M.0
sage: f.prec()
0

sage: M.prec(20)
20
sage: f.prec()
0
sage: x = f.q_expansion() ; f.prec()
20

Python

>>> from sage.all import *
>>> M = ModularForms(Integer(2),Integer(14))
>>> f = M.gen(0)
>>> f.prec()
0

>>> M.prec(Integer(20))
20
>>> f.prec()
0
>>> x = f.q_expansion() ; f.prec()
20

Sage Live

M = ModularForms(2,14)
f = M.0
f.prec()
M.prec(20)
f.prec()
x = f.q_expansion() ; f.prec()

q_expansion(prec=None)

The \(q\)-expansion of the modular form to precision \(O(q^\text{prec})\). This function takes one argument, which is the integer prec.

EXAMPLES:

We compute the cusp form \(\Delta\):

Sage

sage: delta = CuspForms(1,12).0
sage: delta.q_expansion()
q - 24*q^2 + 252*q^3 - 1472*q^4 + 4830*q^5 + O(q^6)

Python

>>> from sage.all import *
>>> delta = CuspForms(Integer(1),Integer(12)).gen(0)
>>> delta.q_expansion()
q - 24*q^2 + 252*q^3 - 1472*q^4 + 4830*q^5 + O(q^6)

Sage Live

delta = CuspForms(1,12).0
delta.q_expansion()

We compute the \(q\)-expansion of one of the cusp forms of level 23:

Sage

sage: f = CuspForms(23,2).0
sage: f.q_expansion()
q - q^3 - q^4 + O(q^6)
sage: f.q_expansion(10)
q - q^3 - q^4 - 2*q^6 + 2*q^7 - q^8 + 2*q^9 + O(q^10)
sage: f.q_expansion(2)
q + O(q^2)
sage: f.q_expansion(1)
O(q^1)
sage: f.q_expansion(0)
O(q^0)
sage: f.q_expansion(-1)
Traceback (most recent call last):
...
ValueError: prec (= -1) must be nonnegative

Python

>>> from sage.all import *
>>> f = CuspForms(Integer(23),Integer(2)).gen(0)
>>> f.q_expansion()
q - q^3 - q^4 + O(q^6)
>>> f.q_expansion(Integer(10))
q - q^3 - q^4 - 2*q^6 + 2*q^7 - q^8 + 2*q^9 + O(q^10)
>>> f.q_expansion(Integer(2))
q + O(q^2)
>>> f.q_expansion(Integer(1))
O(q^1)
>>> f.q_expansion(Integer(0))
O(q^0)
>>> f.q_expansion(-Integer(1))
Traceback (most recent call last):
...
ValueError: prec (= -1) must be nonnegative

Sage Live

f = CuspForms(23,2).0
f.q_expansion()
f.q_expansion(10)
f.q_expansion(2)
f.q_expansion(1)
f.q_expansion(0)
f.q_expansion(-1)

qexp(prec=None)

Same as self.q_expansion(prec).

See also

q_expansion()

EXAMPLES:

Sage

sage: CuspForms(1,12).0.qexp()
q - 24*q^2 + 252*q^3 - 1472*q^4 + 4830*q^5 + O(q^6)

Python

>>> from sage.all import *
>>> CuspForms(Integer(1),Integer(12)).gen(0).qexp()
q - 24*q^2 + 252*q^3 - 1472*q^4 + 4830*q^5 + O(q^6)

Sage Live

CuspForms(1,12).0.qexp()

serre_derivative()

Return the Serre derivative of the given modular form.

If self is of weight \(k\), then the returned modular form will be of weight \(k+2\).

EXAMPLES:

Sage

sage: E4 = ModularForms(1, 4).0
sage: E6 = ModularForms(1, 6).0
sage: DE4 = E4.serre_derivative(); DE4
-1/3 + 168*q + 5544*q^2 + 40992*q^3 + 177576*q^4 + 525168*q^5 + O(q^6)
sage: DE6 = E6.serre_derivative(); DE6
-1/2 - 240*q - 30960*q^2 - 525120*q^3 - 3963120*q^4 - 18750240*q^5 + O(q^6)
sage: Del = ModularForms(1, 12).0 # Modular discriminant
sage: Del.serre_derivative()
0
sage: f = ModularForms(DirichletGroup(5).0, 1).0
sage: Df = f.serre_derivative(); Df
-1/12 + (-11/12*zeta4 + 19/4)*q + (11/6*zeta4 + 59/3)*q^2 + (-41/3*zeta4 + 239/6)*q^3 + (31/4*zeta4 + 839/12)*q^4 + (-251/12*zeta4 + 459/4)*q^5 + O(q^6)

Python

>>> from sage.all import *
>>> E4 = ModularForms(Integer(1), Integer(4)).gen(0)
>>> E6 = ModularForms(Integer(1), Integer(6)).gen(0)
>>> DE4 = E4.serre_derivative(); DE4
-1/3 + 168*q + 5544*q^2 + 40992*q^3 + 177576*q^4 + 525168*q^5 + O(q^6)
>>> DE6 = E6.serre_derivative(); DE6
-1/2 - 240*q - 30960*q^2 - 525120*q^3 - 3963120*q^4 - 18750240*q^5 + O(q^6)
>>> Del = ModularForms(Integer(1), Integer(12)).gen(0) # Modular discriminant
>>> Del.serre_derivative()
0
>>> f = ModularForms(DirichletGroup(Integer(5)).gen(0), Integer(1)).gen(0)
>>> Df = f.serre_derivative(); Df
-1/12 + (-11/12*zeta4 + 19/4)*q + (11/6*zeta4 + 59/3)*q^2 + (-41/3*zeta4 + 239/6)*q^3 + (31/4*zeta4 + 839/12)*q^4 + (-251/12*zeta4 + 459/4)*q^5 + O(q^6)

Sage Live

E4 = ModularForms(1, 4).0
E6 = ModularForms(1, 6).0
DE4 = E4.serre_derivative(); DE4
DE6 = E6.serre_derivative(); DE6
Del = ModularForms(1, 12).0 # Modular discriminant
Del.serre_derivative()
f = ModularForms(DirichletGroup(5).0, 1).0
Df = f.serre_derivative(); Df

The Serre derivative raises the weight of a modular form by \(2\):

Sage

sage: DE4.weight()
6
sage: DE6.weight()
8
sage: Df.weight()
3

Python

>>> from sage.all import *
>>> DE4.weight()
6
>>> DE6.weight()
8
>>> Df.weight()
3

Sage Live

DE4.weight()
DE6.weight()
Df.weight()

The Ramanujan identities are verified (see Wikipedia article Eisenstein_series#Ramanujan_identities):

Sage

sage: DE4 == (-1/3) * E6
True
sage: DE6 == (-1/2) * E4 * E4
True

Python

>>> from sage.all import *
>>> DE4 == (-Integer(1)/Integer(3)) * E6
True
>>> DE6 == (-Integer(1)/Integer(2)) * E4 * E4
True

Sage Live

DE4 == (-1/3) * E6
DE6 == (-1/2) * E4 * E4

symsquare_lseries(chi=None, embedding=0, prec=53)

Compute the symmetric square \(L\)-series of this modular form, twisted by the character \(\chi\).

INPUT:

• chi – Dirichlet character to twist by, or None (default: None), interpreted as the trivial character)

• embedding – embedding of the coefficient field into \(\RR\) or \(\CC\), or an integer \(i\) (in which case take the \(i\)-th embedding)

• prec – the desired precision in bits (default: 53)

OUTPUT: the symmetric square \(L\)-series of the cusp form, as a sage.lfunctions.dokchitser.Dokchitser object.

EXAMPLES:

Sage

sage: CuspForms(1, 12).0.symsquare_lseries()(22)
0.999645711124771

Python

>>> from sage.all import *
>>> CuspForms(Integer(1), Integer(12)).gen(0).symsquare_lseries()(Integer(22))
0.999645711124771

Sage Live

CuspForms(1, 12).0.symsquare_lseries()(22)

An example twisted by a nontrivial character:

Sage

sage: psi = DirichletGroup(7).0^2
sage: L = CuspForms(1, 16).0.symsquare_lseries(psi)
sage: L(22)
0.998407750967420 - 0.00295712911510708*I

Python

>>> from sage.all import *
>>> psi = DirichletGroup(Integer(7)).gen(0)**Integer(2)
>>> L = CuspForms(Integer(1), Integer(16)).gen(0).symsquare_lseries(psi)
>>> L(Integer(22))
0.998407750967420 - 0.00295712911510708*I

Sage Live

psi = DirichletGroup(7).0^2
L = CuspForms(1, 16).0.symsquare_lseries(psi)
L(22)

An example with coefficients not in \(\QQ\):

Sage

sage: F = Newforms(1, 24, names='a')[0]
sage: K = F.hecke_eigenvalue_field()
sage: phi = K.embeddings(RR)[0]
sage: L = F.symsquare_lseries(embedding=phi)
sage: L(5)
verbose -1 (...: dokchitser.py, __call__) Warning: Loss of 8 decimal digits due to cancellation
-3.57698266793901e19

Python

>>> from sage.all import *
>>> F = Newforms(Integer(1), Integer(24), names='a')[Integer(0)]
>>> K = F.hecke_eigenvalue_field()
>>> phi = K.embeddings(RR)[Integer(0)]
>>> L = F.symsquare_lseries(embedding=phi)
>>> L(Integer(5))
verbose -1 (...: dokchitser.py, __call__) Warning: Loss of 8 decimal digits due to cancellation
-3.57698266793901e19

Sage Live

F = Newforms(1, 24, names='a')[0]
K = F.hecke_eigenvalue_field()
phi = K.embeddings(RR)[0]
L = F.symsquare_lseries(embedding=phi)
L(5)

AUTHORS:

• Martin Raum (2011) – original code posted to sage-nt

• David Loeffler (2015) – added support for twists, integrated into Sage library

valuation()

Return the valuation of self (i.e. as an element of the power series ring in q).

EXAMPLES:

Sage

sage: ModularForms(11,2).0.valuation()
1
sage: ModularForms(11,2).1.valuation()
0
sage: ModularForms(25,6).1.valuation()
2
sage: ModularForms(25,6).6.valuation()
7

Python

>>> from sage.all import *
>>> ModularForms(Integer(11),Integer(2)).gen(0).valuation()
1
>>> ModularForms(Integer(11),Integer(2)).gen(1).valuation()
0
>>> ModularForms(Integer(25),Integer(6)).gen(1).valuation()
2
>>> ModularForms(Integer(25),Integer(6)).gen(6).valuation()
7

Sage Live

ModularForms(11,2).0.valuation()
ModularForms(11,2).1.valuation()
ModularForms(25,6).1.valuation()
ModularForms(25,6).6.valuation()

weight()

Return the weight of self.

EXAMPLES:

Sage

sage: (ModularForms(Gamma1(9),2).6).weight()
2

Python

>>> from sage.all import *
>>> (ModularForms(Gamma1(Integer(9)),Integer(2)).gen(6)).weight()
2

Sage Live

(ModularForms(Gamma1(9),2).6).weight()

class sage.modular.modform.element.Newform(parent, component, names, check=True)

Bases: ModularForm_abstract

Initialize a Newform object.

INPUT:

• parent – an ambient cuspidal space of modular forms for which self is a newform

• component – a simple component of a cuspidal modular symbols space of any sign corresponding to this newform

• check – if check is True, check that parent and component have the same weight, level, and character, that component has sign 1 and is simple, and that the types are correct on all inputs.

EXAMPLES:

Sage

sage: sage.modular.modform.element.Newform(CuspForms(11,2), ModularSymbols(11,2,sign=1).cuspidal_subspace(), 'a')
q - 2*q^2 - q^3 + 2*q^4 + q^5 + O(q^6)

sage: f = Newforms(DirichletGroup(5).0, 7,names='a')[0]; f[2].trace(f.base_ring().base_field())
-5*zeta4 - 5

Python

>>> from sage.all import *
>>> sage.modular.modform.element.Newform(CuspForms(Integer(11),Integer(2)), ModularSymbols(Integer(11),Integer(2),sign=Integer(1)).cuspidal_subspace(), 'a')
q - 2*q^2 - q^3 + 2*q^4 + q^5 + O(q^6)

>>> f = Newforms(DirichletGroup(Integer(5)).gen(0), Integer(7),names='a')[Integer(0)]; f[Integer(2)].trace(f.base_ring().base_field())
-5*zeta4 - 5

Sage Live

sage.modular.modform.element.Newform(CuspForms(11,2), ModularSymbols(11,2,sign=1).cuspidal_subspace(), 'a')
f = Newforms(DirichletGroup(5).0, 7,names='a')[0]; f[2].trace(f.base_ring().base_field())

abelian_variety()

Return the abelian variety associated to self.

EXAMPLES:

Sage

sage: Newforms(14,2)[0]
q - q^2 - 2*q^3 + q^4 + O(q^6)
sage: Newforms(14,2)[0].abelian_variety()
Newform abelian subvariety 14a of dimension 1 of J0(14)
sage: Newforms(1, 12)[0].abelian_variety()
Traceback (most recent call last):
...
TypeError: f must have weight 2

Python

>>> from sage.all import *
>>> Newforms(Integer(14),Integer(2))[Integer(0)]
q - q^2 - 2*q^3 + q^4 + O(q^6)
>>> Newforms(Integer(14),Integer(2))[Integer(0)].abelian_variety()
Newform abelian subvariety 14a of dimension 1 of J0(14)
>>> Newforms(Integer(1), Integer(12))[Integer(0)].abelian_variety()
Traceback (most recent call last):
...
TypeError: f must have weight 2

Sage Live

Newforms(14,2)[0]
Newforms(14,2)[0].abelian_variety()
Newforms(1, 12)[0].abelian_variety()

atkin_lehner_action(d=None, normalization='analytic', embedding=None)

Return the result of the Atkin-Lehner operator \(W_d\) on this form \(f\), in the form of a constant \(\lambda_d(f)\) and a normalized newform \(f'\) such that

\[f \mid W_d = \lambda_d(f) f'.\]

See atkin_lehner_eigenvalue() for further details.

EXAMPLES:

Sage

sage: f = Newforms(DirichletGroup(30).1^2, 2, names='a')[0]
sage: emb = f.base_ring().complex_embeddings()[0]
sage: for d in divisors(30):
....:     print(f.atkin_lehner_action(d, embedding=emb))
(1.00000000000000, q + a0*q^2 - a0*q^3 - q^4 + (a0 - 2)*q^5 + O(q^6))
(-1.00000000000000*I, q + a0*q^2 - a0*q^3 - q^4 + (a0 - 2)*q^5 + O(q^6))
(1.00000000000000*I, q + a0*q^2 - a0*q^3 - q^4 + (a0 - 2)*q^5 + O(q^6))
(-0.894427190999916 + 0.447213595499958*I, q - a0*q^2 + a0*q^3 - q^4 + (-a0 - 2)*q^5 + O(q^6))
(1.00000000000000, q + a0*q^2 - a0*q^3 - q^4 + (a0 - 2)*q^5 + O(q^6))
(-0.447213595499958 - 0.894427190999916*I, q - a0*q^2 + a0*q^3 - q^4 + (-a0 - 2)*q^5 + O(q^6))
(0.447213595499958 + 0.894427190999916*I, q - a0*q^2 + a0*q^3 - q^4 + (-a0 - 2)*q^5 + O(q^6))
(-0.894427190999916 + 0.447213595499958*I, q - a0*q^2 + a0*q^3 - q^4 + (-a0 - 2)*q^5 + O(q^6))

Python

>>> from sage.all import *
>>> f = Newforms(DirichletGroup(Integer(30)).gen(1)**Integer(2), Integer(2), names='a')[Integer(0)]
>>> emb = f.base_ring().complex_embeddings()[Integer(0)]
>>> for d in divisors(Integer(30)):
...     print(f.atkin_lehner_action(d, embedding=emb))
(1.00000000000000, q + a0*q^2 - a0*q^3 - q^4 + (a0 - 2)*q^5 + O(q^6))
(-1.00000000000000*I, q + a0*q^2 - a0*q^3 - q^4 + (a0 - 2)*q^5 + O(q^6))
(1.00000000000000*I, q + a0*q^2 - a0*q^3 - q^4 + (a0 - 2)*q^5 + O(q^6))
(-0.894427190999916 + 0.447213595499958*I, q - a0*q^2 + a0*q^3 - q^4 + (-a0 - 2)*q^5 + O(q^6))
(1.00000000000000, q + a0*q^2 - a0*q^3 - q^4 + (a0 - 2)*q^5 + O(q^6))
(-0.447213595499958 - 0.894427190999916*I, q - a0*q^2 + a0*q^3 - q^4 + (-a0 - 2)*q^5 + O(q^6))
(0.447213595499958 + 0.894427190999916*I, q - a0*q^2 + a0*q^3 - q^4 + (-a0 - 2)*q^5 + O(q^6))
(-0.894427190999916 + 0.447213595499958*I, q - a0*q^2 + a0*q^3 - q^4 + (-a0 - 2)*q^5 + O(q^6))

Sage Live

f = Newforms(DirichletGroup(30).1^2, 2, names='a')[0]
emb = f.base_ring().complex_embeddings()[0]
for d in divisors(30):
    print(f.atkin_lehner_action(d, embedding=emb))

The above computation can also be done exactly:

Sage

sage: K.<z> = CyclotomicField(20)
sage: f = Newforms(DirichletGroup(30).1^2, 2, names='a')[0]
sage: emb = f.base_ring().embeddings(CyclotomicField(20, 'z'))[0]
sage: for d in divisors(30):
....:     print(f.atkin_lehner_action(d, embedding=emb))
(1, q + a0*q^2 - a0*q^3 - q^4 + (a0 - 2)*q^5 + O(q^6))
(z^5, q + a0*q^2 - a0*q^3 - q^4 + (a0 - 2)*q^5 + O(q^6))
(-z^5, q + a0*q^2 - a0*q^3 - q^4 + (a0 - 2)*q^5 + O(q^6))
(-2/5*z^7 + 4/5*z^6 + 1/5*z^5 - 4/5*z^4 - 2/5*z^3 - 2/5, q - a0*q^2 + a0*q^3 - q^4 + (-a0 - 2)*q^5 + O(q^6))
(1, q + a0*q^2 - a0*q^3 - q^4 + (a0 - 2)*q^5 + O(q^6))
(4/5*z^7 + 2/5*z^6 - 2/5*z^5 - 2/5*z^4 + 4/5*z^3 - 1/5, q - a0*q^2 + a0*q^3 - q^4 + (-a0 - 2)*q^5 + O(q^6))
(-4/5*z^7 - 2/5*z^6 + 2/5*z^5 + 2/5*z^4 - 4/5*z^3 + 1/5, q - a0*q^2 + a0*q^3 - q^4 + (-a0 - 2)*q^5 + O(q^6))
(-2/5*z^7 + 4/5*z^6 + 1/5*z^5 - 4/5*z^4 - 2/5*z^3 - 2/5, q - a0*q^2 + a0*q^3 - q^4 + (-a0 - 2)*q^5 + O(q^6))

Python

>>> from sage.all import *
>>> K = CyclotomicField(Integer(20), names=('z',)); (z,) = K._first_ngens(1)
>>> f = Newforms(DirichletGroup(Integer(30)).gen(1)**Integer(2), Integer(2), names='a')[Integer(0)]
>>> emb = f.base_ring().embeddings(CyclotomicField(Integer(20), 'z'))[Integer(0)]
>>> for d in divisors(Integer(30)):
...     print(f.atkin_lehner_action(d, embedding=emb))
(1, q + a0*q^2 - a0*q^3 - q^4 + (a0 - 2)*q^5 + O(q^6))
(z^5, q + a0*q^2 - a0*q^3 - q^4 + (a0 - 2)*q^5 + O(q^6))
(-z^5, q + a0*q^2 - a0*q^3 - q^4 + (a0 - 2)*q^5 + O(q^6))
(-2/5*z^7 + 4/5*z^6 + 1/5*z^5 - 4/5*z^4 - 2/5*z^3 - 2/5, q - a0*q^2 + a0*q^3 - q^4 + (-a0 - 2)*q^5 + O(q^6))
(1, q + a0*q^2 - a0*q^3 - q^4 + (a0 - 2)*q^5 + O(q^6))
(4/5*z^7 + 2/5*z^6 - 2/5*z^5 - 2/5*z^4 + 4/5*z^3 - 1/5, q - a0*q^2 + a0*q^3 - q^4 + (-a0 - 2)*q^5 + O(q^6))
(-4/5*z^7 - 2/5*z^6 + 2/5*z^5 + 2/5*z^4 - 4/5*z^3 + 1/5, q - a0*q^2 + a0*q^3 - q^4 + (-a0 - 2)*q^5 + O(q^6))
(-2/5*z^7 + 4/5*z^6 + 1/5*z^5 - 4/5*z^4 - 2/5*z^3 - 2/5, q - a0*q^2 + a0*q^3 - q^4 + (-a0 - 2)*q^5 + O(q^6))

Sage Live

K.<z> = CyclotomicField(20)
f = Newforms(DirichletGroup(30).1^2, 2, names='a')[0]
emb = f.base_ring().embeddings(CyclotomicField(20, 'z'))[0]
for d in divisors(30):
    print(f.atkin_lehner_action(d, embedding=emb))

We can compute the eigenvalue of \(W_{p^e}\) in certain cases where the \(p\)-th coefficient of \(f\) is zero:

Sage

sage: f = Newforms(169, names='a')[0]; f
q + a0*q^2 + 2*q^3 + q^4 - a0*q^5 + O(q^6)
sage: f[13]
0
sage: f.atkin_lehner_eigenvalue(169)
-1

Python

>>> from sage.all import *
>>> f = Newforms(Integer(169), names='a')[Integer(0)]; f
q + a0*q^2 + 2*q^3 + q^4 - a0*q^5 + O(q^6)
>>> f[Integer(13)]
0
>>> f.atkin_lehner_eigenvalue(Integer(169))
-1

Sage Live

f = Newforms(169, names='a')[0]; f
f[13]
f.atkin_lehner_eigenvalue(169)

An example showing the non-multiplicativity of the pseudo-eigenvalues:

Sage

sage: chi = DirichletGroup(18).0^4
sage: f = Newforms(chi, 2)[0]
sage: w2, _ = f.atkin_lehner_action(2); w2
zeta6
sage: w9, _ = f.atkin_lehner_action(9); w9
-zeta18^4
sage: w18,_ = f.atkin_lehner_action(18); w18
-zeta18
sage: w18 == w2 * w9 * chi( crt(2, 9, 9, 2) )
True

Python

>>> from sage.all import *
>>> chi = DirichletGroup(Integer(18)).gen(0)**Integer(4)
>>> f = Newforms(chi, Integer(2))[Integer(0)]
>>> w2, _ = f.atkin_lehner_action(Integer(2)); w2
zeta6
>>> w9, _ = f.atkin_lehner_action(Integer(9)); w9
-zeta18^4
>>> w18,_ = f.atkin_lehner_action(Integer(18)); w18
-zeta18
>>> w18 == w2 * w9 * chi( crt(Integer(2), Integer(9), Integer(9), Integer(2)) )
True

Sage Live

chi = DirichletGroup(18).0^4
f = Newforms(chi, 2)[0]
w2, _ = f.atkin_lehner_action(2); w2
w9, _ = f.atkin_lehner_action(9); w9
w18,_ = f.atkin_lehner_action(18); w18
w18 == w2 * w9 * chi( crt(2, 9, 9, 2) )

atkin_lehner_eigenvalue(d=None, normalization='analytic', embedding=None)

Return the pseudo-eigenvalue of the Atkin-Lehner operator \(W_d\) acting on this form \(f\).

INPUT:

• d – positive integer exactly dividing the level \(N\) of \(f\), i.e., \(d\) divides \(N\) and is coprime to \(N/d\); the default is \(d = N\)

  If \(d\) does not divide \(N\) exactly, then it will be replaced with a multiple \(D\) of \(d\) such that \(D\) exactly divides \(N\) and \(D\) has the same prime factors as \(d\). An error will be raised if \(d\) does not divide \(N\).

• normalization – either 'analytic' (the default) or 'arithmetic'; see below

• embedding – (optional) embedding of the coefficient field of \(f\) into another ring; ignored if 'normalization='arithmetic'

OUTPUT:

The Atkin-Lehner pseudo-eigenvalue of \(W_d\) on \(f\), as an element of the coefficient field of \(f\), or the codomain of embedding if specified.

As defined in [AL1978], the pseudo-eigenvalue is the constant \(\lambda_d(f)\) such that

..math:

f \mid W_d = \lambda_d(f) f'

where \(f'\) is some normalised newform (not necessarily equal to \(f\)).

If normalisation='analytic' (the default), this routine will compute \(\lambda_d\), using the conventions of [AL1978] for the weight \(k\) action, which imply that \(\lambda_d\) has complex absolute value 1. However, with these conventions \(\lambda_d\) is not in the Hecke eigenvalue field of \(f\) in general, so it is often necessary to specify an embedding of the eigenvalue field into a larger ring (which needs to contain roots of unity of sufficiently large order, and a square root of \(d\) if \(k\) is odd).

If normalisation='arithmetic' we compute instead the quotient

..math:

d^{k/2-1} \lambda_d(f) \varepsilon_{N/d}(d / d_0) / G(\varepsilon_d),

where \(G(\varepsilon_d)\) is the Gauss sum of the \(d\)-primary part of the nebentype of \(f\) (more precisely, of its associated primitive character), and \(d_0\) its conductor. This ratio is always in the Hecke eigenvalue field of \(f\) (and can be computed using only arithmetic in this field), so specifying an embedding is not needed, although we still allow it for consistency.

(Note that if \(k = 2\) and \(\varepsilon\) is trivial, both normalisations coincide.)

See also

• sage.modular.hecke.module.atkin_lehner_operator() (especially for the conventions used to define the operator \(W_d\))

• atkin_lehner_action(), which returns both the pseudo-eigenvalue and the newform \(f'\).

EXAMPLES:

Sage

sage: [x.atkin_lehner_eigenvalue() for x in ModularForms(53).newforms('a')]
[1, -1]

sage: f = Newforms(Gamma1(15), 3, names='a')[2]; f
q + a2*q^2 + (-a2 - 2)*q^3 - q^4 - a2*q^5 + O(q^6)
sage: f.atkin_lehner_eigenvalue(5)
Traceback (most recent call last):
...
ValueError: Unable to compute square root. Try specifying an embedding into a larger ring
sage: L = f.hecke_eigenvalue_field(); x = polygen(QQ); M.<sqrt5> = L.extension(x^2 - 5)
sage: f.atkin_lehner_eigenvalue(5, embedding=M.coerce_map_from(L))
1/5*a2*sqrt5
sage: f.atkin_lehner_eigenvalue(5, normalization='arithmetic')
a2

sage: Newforms(DirichletGroup(5).0^2, 6, names='a')[0].atkin_lehner_eigenvalue()
Traceback (most recent call last):
...
ValueError: Unable to compute Gauss sum. Try specifying an embedding into a larger ring

Python

>>> from sage.all import *
>>> [x.atkin_lehner_eigenvalue() for x in ModularForms(Integer(53)).newforms('a')]
[1, -1]

>>> f = Newforms(Gamma1(Integer(15)), Integer(3), names='a')[Integer(2)]; f
q + a2*q^2 + (-a2 - 2)*q^3 - q^4 - a2*q^5 + O(q^6)
>>> f.atkin_lehner_eigenvalue(Integer(5))
Traceback (most recent call last):
...
ValueError: Unable to compute square root. Try specifying an embedding into a larger ring
>>> L = f.hecke_eigenvalue_field(); x = polygen(QQ); M = L.extension(x**Integer(2) - Integer(5), names=('sqrt5',)); (sqrt5,) = M._first_ngens(1)
>>> f.atkin_lehner_eigenvalue(Integer(5), embedding=M.coerce_map_from(L))
1/5*a2*sqrt5
>>> f.atkin_lehner_eigenvalue(Integer(5), normalization='arithmetic')
a2

>>> Newforms(DirichletGroup(Integer(5)).gen(0)**Integer(2), Integer(6), names='a')[Integer(0)].atkin_lehner_eigenvalue()
Traceback (most recent call last):
...
ValueError: Unable to compute Gauss sum. Try specifying an embedding into a larger ring

Sage Live

[x.atkin_lehner_eigenvalue() for x in ModularForms(53).newforms('a')]
f = Newforms(Gamma1(15), 3, names='a')[2]; f
f.atkin_lehner_eigenvalue(5)
L = f.hecke_eigenvalue_field(); x = polygen(QQ); M.<sqrt5> = L.extension(x^2 - 5)
f.atkin_lehner_eigenvalue(5, embedding=M.coerce_map_from(L))
f.atkin_lehner_eigenvalue(5, normalization='arithmetic')
Newforms(DirichletGroup(5).0^2, 6, names='a')[0].atkin_lehner_eigenvalue()

character()

The nebentypus character of this newform (as a Dirichlet character with values in the field of Hecke eigenvalues of the form).

EXAMPLES:

Sage

sage: Newforms(Gamma1(7), 4,names='a')[1].character()
Dirichlet character modulo 7 of conductor 7 mapping 3 |--> 1/2*a1
sage: chi = DirichletGroup(3).0; Newforms(chi, 7)[0].character() == chi
True

Python

>>> from sage.all import *
>>> Newforms(Gamma1(Integer(7)), Integer(4),names='a')[Integer(1)].character()
Dirichlet character modulo 7 of conductor 7 mapping 3 |--> 1/2*a1
>>> chi = DirichletGroup(Integer(3)).gen(0); Newforms(chi, Integer(7))[Integer(0)].character() == chi
True

Sage Live

Newforms(Gamma1(7), 4,names='a')[1].character()
chi = DirichletGroup(3).0; Newforms(chi, 7)[0].character() == chi

coefficient(n)

Return the coefficient of \(q^n\) in the power series of self.

INPUT:

• n – positive integer

OUTPUT: the coefficient of \(q^n\) in the power series of self

EXAMPLES:

Sage

sage: f = Newforms(11)[0]; f
q - 2*q^2 - q^3 + 2*q^4 + q^5 + O(q^6)
sage: f.coefficient(100)
-8

sage: g = Newforms(23, names='a')[0]; g
q + a0*q^2 + (-2*a0 - 1)*q^3 + (-a0 - 1)*q^4 + 2*a0*q^5 + O(q^6)
sage: g.coefficient(3)
-2*a0 - 1

Python

>>> from sage.all import *
>>> f = Newforms(Integer(11))[Integer(0)]; f
q - 2*q^2 - q^3 + 2*q^4 + q^5 + O(q^6)
>>> f.coefficient(Integer(100))
-8

>>> g = Newforms(Integer(23), names='a')[Integer(0)]; g
q + a0*q^2 + (-2*a0 - 1)*q^3 + (-a0 - 1)*q^4 + 2*a0*q^5 + O(q^6)
>>> g.coefficient(Integer(3))
-2*a0 - 1

Sage Live

f = Newforms(11)[0]; f
f.coefficient(100)
g = Newforms(23, names='a')[0]; g
g.coefficient(3)

element()

Find an element of the ambient space of modular forms which represents this newform.

Note

This can be quite expensive. Also, the polynomial defining the field of Hecke eigenvalues should be considered random, since it is generated by a random sum of Hecke operators. (The field itself is not random, of course.)

EXAMPLES:

Sage

sage: ls = Newforms(38,4,names='a')
sage: ls[0]
q - 2*q^2 - 2*q^3 + 4*q^4 - 9*q^5 + O(q^6)
sage: ls # random
[q - 2*q^2 - 2*q^3 + 4*q^4 - 9*q^5 + O(q^6),
q - 2*q^2 + (-a1 - 2)*q^3 + 4*q^4 + (2*a1 + 10)*q^5 + O(q^6),
q + 2*q^2 + (1/2*a2 - 1)*q^3 + 4*q^4 + (-3/2*a2 + 12)*q^5 + O(q^6)]
sage: type(ls[0])
<class 'sage.modular.modform.element.Newform'>
sage: ls[2][3].minpoly()
x^2 - 9*x + 2
sage: ls2 = [ x.element() for x in ls ]
sage: ls2 # random
[q - 2*q^2 - 2*q^3 + 4*q^4 - 9*q^5 + O(q^6),
q - 2*q^2 + (-a1 - 2)*q^3 + 4*q^4 + (2*a1 + 10)*q^5 + O(q^6),
q + 2*q^2 + (1/2*a2 - 1)*q^3 + 4*q^4 + (-3/2*a2 + 12)*q^5 + O(q^6)]
sage: type(ls2[0])
<class 'sage.modular.modform.cuspidal_submodule.CuspidalSubmodule_g0_Q_with_category.element_class'>
sage: ls2[2][3].minpoly()
x^2 - 9*x + 2

Python

>>> from sage.all import *
>>> ls = Newforms(Integer(38),Integer(4),names='a')
>>> ls[Integer(0)]
q - 2*q^2 - 2*q^3 + 4*q^4 - 9*q^5 + O(q^6)
>>> ls # random
[q - 2*q^2 - 2*q^3 + 4*q^4 - 9*q^5 + O(q^6),
q - 2*q^2 + (-a1 - 2)*q^3 + 4*q^4 + (2*a1 + 10)*q^5 + O(q^6),
q + 2*q^2 + (1/2*a2 - 1)*q^3 + 4*q^4 + (-3/2*a2 + 12)*q^5 + O(q^6)]
>>> type(ls[Integer(0)])
<class 'sage.modular.modform.element.Newform'>
>>> ls[Integer(2)][Integer(3)].minpoly()
x^2 - 9*x + 2
>>> ls2 = [ x.element() for x in ls ]
>>> ls2 # random
[q - 2*q^2 - 2*q^3 + 4*q^4 - 9*q^5 + O(q^6),
q - 2*q^2 + (-a1 - 2)*q^3 + 4*q^4 + (2*a1 + 10)*q^5 + O(q^6),
q + 2*q^2 + (1/2*a2 - 1)*q^3 + 4*q^4 + (-3/2*a2 + 12)*q^5 + O(q^6)]
>>> type(ls2[Integer(0)])
<class 'sage.modular.modform.cuspidal_submodule.CuspidalSubmodule_g0_Q_with_category.element_class'>
>>> ls2[Integer(2)][Integer(3)].minpoly()
x^2 - 9*x + 2

Sage Live

ls = Newforms(38,4,names='a')
ls[0]
ls # random
type(ls[0])
ls[2][3].minpoly()
ls2 = [ x.element() for x in ls ]
ls2 # random
type(ls2[0])
ls2[2][3].minpoly()

hecke_eigenvalue_field()

Return the field generated over the rationals by the coefficients of this newform.

EXAMPLES:

Sage

sage: ls = Newforms(35, 2, names='a') ; ls
[q + q^3 - 2*q^4 - q^5 + O(q^6),
q + a1*q^2 + (-a1 - 1)*q^3 + (-a1 + 2)*q^4 + q^5 + O(q^6)]
sage: ls[0].hecke_eigenvalue_field()
Rational Field
sage: ls[1].hecke_eigenvalue_field()
Number Field in a1 with defining polynomial x^2 + x - 4

Python

>>> from sage.all import *
>>> ls = Newforms(Integer(35), Integer(2), names='a') ; ls
[q + q^3 - 2*q^4 - q^5 + O(q^6),
q + a1*q^2 + (-a1 - 1)*q^3 + (-a1 + 2)*q^4 + q^5 + O(q^6)]
>>> ls[Integer(0)].hecke_eigenvalue_field()
Rational Field
>>> ls[Integer(1)].hecke_eigenvalue_field()
Number Field in a1 with defining polynomial x^2 + x - 4

Sage Live

ls = Newforms(35, 2, names='a') ; ls
ls[0].hecke_eigenvalue_field()
ls[1].hecke_eigenvalue_field()

is_cuspidal()

Return True. For compatibility with elements of modular forms spaces.

EXAMPLES:

Sage

sage: Newforms(11, 2)[0].is_cuspidal()
True

Python

>>> from sage.all import *
>>> Newforms(Integer(11), Integer(2))[Integer(0)].is_cuspidal()
True

Sage Live

Newforms(11, 2)[0].is_cuspidal()

local_component(p, twist_factor=None)

Calculate the local component at the prime \(p\) of the automorphic representation attached to this newform. For more information, see the documentation of the LocalComponent() function.

EXAMPLES:

Sage

sage: f = Newform("49a")
sage: f.local_component(7)
Smooth representation of GL_2(Q_7) with conductor 7^2

Python

>>> from sage.all import *
>>> f = Newform("49a")
>>> f.local_component(Integer(7))
Smooth representation of GL_2(Q_7) with conductor 7^2

Sage Live

f = Newform("49a")
f.local_component(7)

minimal_twist(p=None)

Compute a pair \((g, chi)\) such that \(g = f \otimes \chi\), where \(f\) is this newform and \(\chi\) is a Dirichlet character, such that \(g\) has level as small as possible. If the optional argument \(p\) is given, consider only twists by Dirichlet characters of \(p\)-power conductor.

EXAMPLES:

Sage

sage: f = Newforms(121, 2)[3]
sage: g, chi = f.minimal_twist()
sage: g
q - 2*q^2 - q^3 + 2*q^4 + q^5 + O(q^6)
sage: chi
Dirichlet character modulo 11 of conductor 11 mapping 2 |--> -1
sage: f.twist(chi, level=11) == g
True

sage: # long time
sage: f = Newforms(575, 2, names='a')[4]
sage: g, chi = f.minimal_twist(5)
sage: g
q + a*q^2 - a*q^3 - 2*q^4 + (1/2*a + 2)*q^5 + O(q^6)
sage: chi
Dirichlet character modulo 5 of conductor 5 mapping 2 |--> 1/2*a
sage: f.twist(chi, level=g.level()) == g
True

Python

>>> from sage.all import *
>>> f = Newforms(Integer(121), Integer(2))[Integer(3)]
>>> g, chi = f.minimal_twist()
>>> g
q - 2*q^2 - q^3 + 2*q^4 + q^5 + O(q^6)
>>> chi
Dirichlet character modulo 11 of conductor 11 mapping 2 |--> -1
>>> f.twist(chi, level=Integer(11)) == g
True

>>> # long time
>>> f = Newforms(Integer(575), Integer(2), names='a')[Integer(4)]
>>> g, chi = f.minimal_twist(Integer(5))
>>> g
q + a*q^2 - a*q^3 - 2*q^4 + (1/2*a + 2)*q^5 + O(q^6)
>>> chi
Dirichlet character modulo 5 of conductor 5 mapping 2 |--> 1/2*a
>>> f.twist(chi, level=g.level()) == g
True

Sage Live

f = Newforms(121, 2)[3]
g, chi = f.minimal_twist()
g
chi
f.twist(chi, level=11) == g
# long time
f = Newforms(575, 2, names='a')[4]
g, chi = f.minimal_twist(5)
g
chi
f.twist(chi, level=g.level()) == g

modsym_eigenspace(sign=0)

Return a submodule of dimension 1 or 2 of the ambient space of the sign 0 modular symbols space associated to self, base-extended to the Hecke eigenvalue field, which is an eigenspace for the Hecke operators with the same eigenvalues as this newform, and is an eigenspace for the star involution of the appropriate sign if the sign is not 0.

EXAMPLES:

Sage

sage: N = Newform("37a")
sage: N.modular_symbols(0)
Modular Symbols subspace of dimension 2 of Modular Symbols space of dimension 5 for Gamma_0(37) of weight 2 with sign 0 over Rational Field
sage: M = N.modular_symbols(0)
sage: V = N.modsym_eigenspace(1); V
Vector space of degree 5 and dimension 1 over Rational Field
Basis matrix:
[ 0  1 -1  1  0]
sage: V.0 in M.free_module()
True
sage: V = N.modsym_eigenspace(-1); V
Vector space of degree 5 and dimension 1 over Rational Field
Basis matrix:
[   0    0    0    1 -1/2]
sage: V.0 in M.free_module()
True

Python

>>> from sage.all import *
>>> N = Newform("37a")
>>> N.modular_symbols(Integer(0))
Modular Symbols subspace of dimension 2 of Modular Symbols space of dimension 5 for Gamma_0(37) of weight 2 with sign 0 over Rational Field
>>> M = N.modular_symbols(Integer(0))
>>> V = N.modsym_eigenspace(Integer(1)); V
Vector space of degree 5 and dimension 1 over Rational Field
Basis matrix:
[ 0  1 -1  1  0]
>>> V.gen(0) in M.free_module()
True
>>> V = N.modsym_eigenspace(-Integer(1)); V
Vector space of degree 5 and dimension 1 over Rational Field
Basis matrix:
[   0    0    0    1 -1/2]
>>> V.gen(0) in M.free_module()
True

Sage Live

N = Newform("37a")
N.modular_symbols(0)
M = N.modular_symbols(0)
V = N.modsym_eigenspace(1); V
V.0 in M.free_module()
V = N.modsym_eigenspace(-1); V
V.0 in M.free_module()

modular_symbols(sign=0)

Return the subspace with the specified sign of the space of modular symbols corresponding to this newform.

EXAMPLES:

Sage

sage: f = Newforms(18,4)[0]
sage: f.modular_symbols()
Modular Symbols subspace of dimension 2 of Modular Symbols space of dimension 18 for Gamma_0(18) of weight 4 with sign 0 over Rational Field
sage: f.modular_symbols(1)
Modular Symbols subspace of dimension 1 of Modular Symbols space of dimension 11 for Gamma_0(18) of weight 4 with sign 1 over Rational Field

Python

>>> from sage.all import *
>>> f = Newforms(Integer(18),Integer(4))[Integer(0)]
>>> f.modular_symbols()
Modular Symbols subspace of dimension 2 of Modular Symbols space of dimension 18 for Gamma_0(18) of weight 4 with sign 0 over Rational Field
>>> f.modular_symbols(Integer(1))
Modular Symbols subspace of dimension 1 of Modular Symbols space of dimension 11 for Gamma_0(18) of weight 4 with sign 1 over Rational Field

Sage Live

f = Newforms(18,4)[0]
f.modular_symbols()
f.modular_symbols(1)

number()

Return the index of this space in the list of simple, new, cuspidal subspaces of the full space of modular symbols for this weight and level.

EXAMPLES:

Sage

sage: Newforms(43, 2, names='a')[1].number()
1

Python

>>> from sage.all import *
>>> Newforms(Integer(43), Integer(2), names='a')[Integer(1)].number()
1

Sage Live

Newforms(43, 2, names='a')[1].number()

twist(chi, level=None, check=True)

Return the twist of the newform self by the Dirichlet character chi.

If self is a newform \(f\) with character \(\epsilon\) and \(q\)-expansion

\[f(q) = \sum_{n=1}^\infty a_n q^n,\]

then the twist by \(\chi\) is the unique newform \(f\otimes\chi\) with character \(\epsilon\chi^2\) and \(q\)-expansion

\[(f\otimes\chi)(q) = \sum_{n=1}^\infty b_n q^n\]

satisfying \(b_n = \chi(n) a_n\) for all but finitely many \(n\).

INPUT:

• chi – a Dirichlet character. Note that Sage must be able to determine a common base field into which both the Hecke eigenvalue field of self, and the field of values of chi, can be embedded.

• level – (optional) the level \(N\) of the twisted form. If \(N\) is not given, the algorithm tries to compute \(N\) using [AL1978], Theorem 3.1; if this is not possible, it returns an error. If \(N\) is given but incorrect, i.e. the twisted form does not have level \(N\), then this function will attempt to detect this and return an error, but it may sometimes return an incorrect answer (a newform of level \(N\) whose first few coefficients agree with those of \(f \otimes \chi\)).

• check – (optional) boolean; if True (default), ensure that the space of modular symbols that is computed is genuinely simple and new. This makes it less likely, but not impossible, that a wrong result is returned if an incorrect level is specified.

OUTPUT:

The form \(f\otimes\chi\) as an element of the set of newforms for \(\Gamma_1(N)\) with character \(\epsilon\chi^2\).

EXAMPLES:

Sage

sage: G = DirichletGroup(3, base_ring=QQ)
sage: Delta = Newforms(SL2Z, 12)[0]; Delta
q - 24*q^2 + 252*q^3 - 1472*q^4 + 4830*q^5 + O(q^6)
sage: Delta.twist(G[0]) == Delta
True
sage: Delta.twist(G[1])  # long time (about 5 s)
q + 24*q^2 - 1472*q^4 - 4830*q^5 + O(q^6)

sage: M = CuspForms(Gamma1(13), 2)
sage: f = M.newforms('a')[0]; f
q + a0*q^2 + (-2*a0 - 4)*q^3 + (-a0 - 1)*q^4 + (2*a0 + 3)*q^5 + O(q^6)
sage: f.twist(G[1])
q - a0*q^2 + (-a0 - 1)*q^4 + (-2*a0 - 3)*q^5 + O(q^6)

sage: f = Newforms(Gamma1(30), 2, names='a')[1]; f
q + a1*q^2 - a1*q^3 - q^4 + (a1 - 2)*q^5 + O(q^6)
sage: f.twist(f.character())
Traceback (most recent call last):
...
NotImplementedError: cannot calculate 5-primary part of the level of the twist of q + a1*q^2 - a1*q^3 - q^4 + (a1 - 2)*q^5 + O(q^6) by Dirichlet character modulo 5 of conductor 5 mapping 2 |--> -1
sage: f.twist(f.character(), level=30)
q - a1*q^2 + a1*q^3 - q^4 + (-a1 - 2)*q^5 + O(q^6)

Python

>>> from sage.all import *
>>> G = DirichletGroup(Integer(3), base_ring=QQ)
>>> Delta = Newforms(SL2Z, Integer(12))[Integer(0)]; Delta
q - 24*q^2 + 252*q^3 - 1472*q^4 + 4830*q^5 + O(q^6)
>>> Delta.twist(G[Integer(0)]) == Delta
True
>>> Delta.twist(G[Integer(1)])  # long time (about 5 s)
q + 24*q^2 - 1472*q^4 - 4830*q^5 + O(q^6)

>>> M = CuspForms(Gamma1(Integer(13)), Integer(2))
>>> f = M.newforms('a')[Integer(0)]; f
q + a0*q^2 + (-2*a0 - 4)*q^3 + (-a0 - 1)*q^4 + (2*a0 + 3)*q^5 + O(q^6)
>>> f.twist(G[Integer(1)])
q - a0*q^2 + (-a0 - 1)*q^4 + (-2*a0 - 3)*q^5 + O(q^6)

>>> f = Newforms(Gamma1(Integer(30)), Integer(2), names='a')[Integer(1)]; f
q + a1*q^2 - a1*q^3 - q^4 + (a1 - 2)*q^5 + O(q^6)
>>> f.twist(f.character())
Traceback (most recent call last):
...
NotImplementedError: cannot calculate 5-primary part of the level of the twist of q + a1*q^2 - a1*q^3 - q^4 + (a1 - 2)*q^5 + O(q^6) by Dirichlet character modulo 5 of conductor 5 mapping 2 |--> -1
>>> f.twist(f.character(), level=Integer(30))
q - a1*q^2 + a1*q^3 - q^4 + (-a1 - 2)*q^5 + O(q^6)

Sage Live

G = DirichletGroup(3, base_ring=QQ)
Delta = Newforms(SL2Z, 12)[0]; Delta
Delta.twist(G[0]) == Delta
Delta.twist(G[1])  # long time (about 5 s)
M = CuspForms(Gamma1(13), 2)
f = M.newforms('a')[0]; f
f.twist(G[1])
f = Newforms(Gamma1(30), 2, names='a')[1]; f
f.twist(f.character())
f.twist(f.character(), level=30)

AUTHORS:

• Peter Bruin (April 2015)

sage.modular.modform.element.delta_lseries(prec=53, max_imaginary_part=0, max_asymp_coeffs=40, algorithm=None)

Return the \(L\)-series of the modular form \(\Delta\).

If algorithm is 'gp', this returns an interface to Tim Dokchitser’s program for computing with the \(L\)-series of the modular form \(\Delta\).

If algorithm is 'pari', this returns instead an interface to Pari’s own general implementation of \(L\)-functions.

INPUT:

• prec – integer (bits precision)

• max_imaginary_part – real number

• max_asymp_coeffs – integer

• algorithm – string; 'gp' (default), 'pari'

OUTPUT:

The \(L\)-series of \(\Delta\).

EXAMPLES:

Sage

sage: L = delta_lseries()
sage: L(1)
0.0374412812685155

sage: L = delta_lseries(algorithm='pari')
sage: L(1)
0.0374412812685155

Python

>>> from sage.all import *
>>> L = delta_lseries()
>>> L(Integer(1))
0.0374412812685155

>>> L = delta_lseries(algorithm='pari')
>>> L(Integer(1))
0.0374412812685155

Sage Live

L = delta_lseries()
L(1)
L = delta_lseries(algorithm='pari')
L(1)

sage.modular.modform.element.is_ModularFormElement(x)

Return True if x is a modular form.

EXAMPLES:

Sage

sage: from sage.modular.modform.element import is_ModularFormElement
sage: is_ModularFormElement(5)
doctest:warning...
DeprecationWarning: The function is_ModularFormElement is deprecated;
use 'isinstance(..., ModularFormElement)' instead.
See https://github.com/sagemath/sage/issues/38184 for details.
False
sage: is_ModularFormElement(ModularForms(11).0)
True

Python

>>> from sage.all import *
>>> from sage.modular.modform.element import is_ModularFormElement
>>> is_ModularFormElement(Integer(5))
doctest:warning...
DeprecationWarning: The function is_ModularFormElement is deprecated;
use 'isinstance(..., ModularFormElement)' instead.
See https://github.com/sagemath/sage/issues/38184 for details.
False
>>> is_ModularFormElement(ModularForms(Integer(11)).gen(0))
True

Sage Live

from sage.modular.modform.element import is_ModularFormElement
is_ModularFormElement(5)
is_ModularFormElement(ModularForms(11).0)

Make live