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# Modular Forms for \(\Gamma_1(N)\) and \(\Gamma_H(N)\) over \(\QQ\)

EXAMPLES:

    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, [4])); M
    Modular Forms space of dimension 2 for Congruence Subgroup Gamma_H(11) with H generated by [4] 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)
    ]

TESTS:

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

    sage: m = ModularForms(GammaH(15, [4]), 2)
    sage: loads(dumps(m)) == m
    True

We check that #10453 is fixed:

    sage: CuspForms(Gamma1(11), 2).old_submodule()
    Modular Forms subspace of dimension 0 of Modular Forms space of dimension 10 for Congruence Subgroup Gamma1(11) of weight 2 over Rational Field
    sage: ModularForms(Gamma1(3), 12).old_submodule()
    Modular Forms subspace of dimension 4 of Modular Forms space of dimension 5 for Congruence Subgroup Gamma1(3) of weight 12 over Rational Field

class `sage.modular.modform.ambient_g1.``ModularFormsAmbient_g1_Q`(level, weight)

  Bases: `sage.modular.modform.ambient_g1.ModularFormsAmbient_gH_Q`

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

  `cuspidal_submodule`()

    Return the cuspidal submodule of this modular forms space.

    EXAMPLES:

        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`()

    Return the Eisenstein submodule of this modular forms space.

    EXAMPLES:

        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

class `sage.modular.modform.ambient_g1.``ModularFormsAmbient_gH_Q`(group, weight)

  Bases: `sage.modular.modform.ambient.ModularFormsAmbient`

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

  `cuspidal_submodule`()

    Return the cuspidal submodule of this modular forms space.

    EXAMPLES:

        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

  `eisenstein_submodule`()

    Return the Eisenstein submodule of this modular forms space.

    EXAMPLES:

        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'>

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