Modular forms for \(\Gamma_1(N)\) and \(\Gamma_H(N)\) over \(\QQ\)¶

EXAMPLES:

Sage

sage: M = ModularForms(Gamma1(13),2); M
Modular Forms space of dimension 13 for Congruence Subgroup Gamma1(13) of weight 2 over Rational Field
sage: S = M.cuspidal_submodule(); S
Cuspidal subspace of dimension 2 of Modular Forms space of dimension 13 for Congruence Subgroup Gamma1(13) of weight 2 over Rational Field
sage: S.basis()
[
q - 4*q^3 - q^4 + 3*q^5 + O(q^6),
q^2 - 2*q^3 - q^4 + 2*q^5 + O(q^6)
]

sage: M = ModularForms(GammaH(11, [3])); M
Modular Forms space of dimension 2 for Congruence Subgroup Gamma_H(11) with H generated by [3] of weight 2 over Rational Field
sage: M.q_expansion_basis(8)
[
q - 2*q^2 - q^3 + 2*q^4 + q^5 + 2*q^6 - 2*q^7 + O(q^8),
1 + 12/5*q + 36/5*q^2 + 48/5*q^3 + 84/5*q^4 + 72/5*q^5 + 144/5*q^6 + 96/5*q^7 + O(q^8)
]

Python

>>> from sage.all import *
>>> M = ModularForms(Gamma1(Integer(13)),Integer(2)); M
Modular Forms space of dimension 13 for Congruence Subgroup Gamma1(13) of weight 2 over Rational Field
>>> S = M.cuspidal_submodule(); S
Cuspidal subspace of dimension 2 of Modular Forms space of dimension 13 for Congruence Subgroup Gamma1(13) of weight 2 over Rational Field
>>> S.basis()
[
q - 4*q^3 - q^4 + 3*q^5 + O(q^6),
q^2 - 2*q^3 - q^4 + 2*q^5 + O(q^6)
]

>>> M = ModularForms(GammaH(Integer(11), [Integer(3)])); M
Modular Forms space of dimension 2 for Congruence Subgroup Gamma_H(11) with H generated by [3] of weight 2 over Rational Field
>>> M.q_expansion_basis(Integer(8))
[
q - 2*q^2 - q^3 + 2*q^4 + q^5 + 2*q^6 - 2*q^7 + O(q^8),
1 + 12/5*q + 36/5*q^2 + 48/5*q^3 + 84/5*q^4 + 72/5*q^5 + 144/5*q^6 + 96/5*q^7 + O(q^8)
]

Sage Live

M = ModularForms(Gamma1(13),2); M
S = M.cuspidal_submodule(); S
S.basis()
M = ModularForms(GammaH(11, [3])); M
M.q_expansion_basis(8)

class sage.modular.modform.ambient_g1.ModularFormsAmbient_g1_Q(level, weight, eis_only)[source]¶

Bases: ModularFormsAmbient_gH_Q

A space of modular forms for the group \(\Gamma_1(N)\) over the rational numbers.

cuspidal_submodule()[source]¶

Return the cuspidal submodule of this modular forms space.

EXAMPLES:

Sage

sage: m = ModularForms(Gamma1(17),2); m
Modular Forms space of dimension 20 for Congruence Subgroup Gamma1(17) of weight 2 over Rational Field
sage: m.cuspidal_submodule()
Cuspidal subspace of dimension 5 of Modular Forms space of dimension 20 for Congruence Subgroup Gamma1(17) of weight 2 over Rational Field

Python

>>> from sage.all import *
>>> m = ModularForms(Gamma1(Integer(17)),Integer(2)); m
Modular Forms space of dimension 20 for Congruence Subgroup Gamma1(17) of weight 2 over Rational Field
>>> m.cuspidal_submodule()
Cuspidal subspace of dimension 5 of Modular Forms space of dimension 20 for Congruence Subgroup Gamma1(17) of weight 2 over Rational Field

Sage Live

m = ModularForms(Gamma1(17),2); m
m.cuspidal_submodule()

eisenstein_submodule()[source]¶

Return the Eisenstein submodule of this modular forms space.

EXAMPLES:

Sage

sage: ModularForms(Gamma1(13),2).eisenstein_submodule()
Eisenstein subspace of dimension 11 of Modular Forms space of dimension 13 for Congruence Subgroup Gamma1(13) of weight 2 over Rational Field
sage: ModularForms(Gamma1(13),10).eisenstein_submodule()
Eisenstein subspace of dimension 12 of Modular Forms space of dimension 69 for Congruence Subgroup Gamma1(13) of weight 10 over Rational Field

Python

>>> from sage.all import *
>>> ModularForms(Gamma1(Integer(13)),Integer(2)).eisenstein_submodule()
Eisenstein subspace of dimension 11 of Modular Forms space of dimension 13 for Congruence Subgroup Gamma1(13) of weight 2 over Rational Field
>>> ModularForms(Gamma1(Integer(13)),Integer(10)).eisenstein_submodule()
Eisenstein subspace of dimension 12 of Modular Forms space of dimension 69 for Congruence Subgroup Gamma1(13) of weight 10 over Rational Field

Sage Live

ModularForms(Gamma1(13),2).eisenstein_submodule()
ModularForms(Gamma1(13),10).eisenstein_submodule()

class sage.modular.modform.ambient_g1.ModularFormsAmbient_gH_Q(group, weight, eis_only)[source]¶

Bases: ModularFormsAmbient

A space of modular forms for the group \(\Gamma_H(N)\) over the rational numbers.

cuspidal_submodule()[source]¶

Return the cuspidal submodule of this modular forms space.

EXAMPLES:

Sage

sage: m = ModularForms(GammaH(100, [29]),2); m
Modular Forms space of dimension 48 for Congruence Subgroup Gamma_H(100) with H generated by [29] of weight 2 over Rational Field
sage: m.cuspidal_submodule()
Cuspidal subspace of dimension 13 of Modular Forms space of dimension 48 for Congruence Subgroup Gamma_H(100) with H generated by [29] of weight 2 over Rational Field

Python

>>> from sage.all import *
>>> m = ModularForms(GammaH(Integer(100), [Integer(29)]),Integer(2)); m
Modular Forms space of dimension 48 for Congruence Subgroup Gamma_H(100) with H generated by [29] of weight 2 over Rational Field
>>> m.cuspidal_submodule()
Cuspidal subspace of dimension 13 of Modular Forms space of dimension 48 for Congruence Subgroup Gamma_H(100) with H generated by [29] of weight 2 over Rational Field

Sage Live

m = ModularForms(GammaH(100, [29]),2); m
m.cuspidal_submodule()

eisenstein_submodule()[source]¶

Return the Eisenstein submodule of this modular forms space.

EXAMPLES:

Sage

sage: E = ModularForms(GammaH(100, [29]),3).eisenstein_submodule(); E
Eisenstein subspace of dimension 24 of Modular Forms space of dimension 72 for Congruence Subgroup Gamma_H(100) with H generated by [29] of weight 3 over Rational Field
sage: type(E)
<class 'sage.modular.modform.eisenstein_submodule.EisensteinSubmodule_gH_Q_with_category'>

Python

>>> from sage.all import *
>>> E = ModularForms(GammaH(Integer(100), [Integer(29)]),Integer(3)).eisenstein_submodule(); E
Eisenstein subspace of dimension 24 of Modular Forms space of dimension 72 for Congruence Subgroup Gamma_H(100) with H generated by [29] of weight 3 over Rational Field
>>> type(E)
<class 'sage.modular.modform.eisenstein_submodule.EisensteinSubmodule_gH_Q_with_category'>

Sage Live

E = ModularForms(GammaH(100, [29]),3).eisenstein_submodule(); E
type(E)