42.6 Modular Forms for $ \Gamma_1(N)$ over $ \mathbf{Q}$

Module: sage.modular.modform.ambient_g1

Modular Forms for $ \Gamma_1(N)$ over $ \mathbf{Q}$ .

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)
]

TESTS:

sage: m = ModularForms(Gamma1(20),2)
sage: loads(dumps(m)) == m
True

Class: ModularFormsAmbient_g1_Q

class ModularFormsAmbient_g1_Q
A space of modular forms for the group $ \Gamma_1(N)$ over the rational numbers.
ModularFormsAmbient_g1_Q( self, level, weight)

Create a space of modular forms for $ \Gamma_1(N)$ of integral weight over the rational numbers.

sage: m = ModularForms(Gamma1(100),5); m
Modular Forms space of dimension 1270 for Congruence Subgroup Gamma1(100)
of weight 5 over Rational Field
sage: type(m)
<class 'sage.modular.modform.ambient_g1.ModularFormsAmbient_g1_Q'>

Functions: cuspidal_submodule,$ \,$ eisenstein_submodule

cuspidal_submodule( self)

Return the cuspidal submodule of this modular forms space.

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

eisenstein_submodule( self)

Return the Eisenstein submodule of this modular forms space.

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

Special Functions: __init__

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