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# The Cuspidal Subspace

EXAMPLES:

    sage: S = CuspForms(SL2Z,12); S
    Cuspidal subspace of dimension 1 of Modular Forms space of dimension 2 for
    Modular Group SL(2,Z) of weight 12 over Rational Field
    sage: S.basis()
    [
    q - 24*q^2 + 252*q^3 - 1472*q^4 + 4830*q^5 + O(q^6)
    ]

    sage: S = CuspForms(Gamma0(33),2); S
    Cuspidal subspace of dimension 3 of Modular Forms space of dimension 6 for
    Congruence Subgroup Gamma0(33) of weight 2 over Rational Field
    sage: S.basis()
    [
    q - q^5 + O(q^6),
    q^2 - q^4 - q^5 + O(q^6),
    q^3 + O(q^6)
    ]

    sage: S = CuspForms(Gamma1(3),6); S
    Cuspidal subspace of dimension 1 of Modular Forms space of dimension 3 for
    Congruence Subgroup Gamma1(3) of weight 6 over Rational Field
    sage: S.basis()
    [
    q - 6*q^2 + 9*q^3 + 4*q^4 + 6*q^5 + O(q^6)
    ]

class `sage.modular.modform.cuspidal_submodule.``CuspidalSubmodule`(ambient_space)

  Bases: `sage.modular.modform.submodule.ModularFormsSubmodule`

  Base class for cuspidal submodules of ambient spaces of modular forms.

  `change_ring`(R)

    Change the base ring of self to R, when this makes sense. This differs from `base_extend()` in that there may not be a canonical map from self to the new space, as in the first example below. If this space has a character then this may fail when the character cannot be defined over R, as in the second example.

    EXAMPLES:

        sage: chi = DirichletGroup(109, CyclotomicField(3)).0
        sage: S9 = CuspForms(chi, 2, base_ring = CyclotomicField(9)); S9
        Cuspidal subspace of dimension 8 of Modular Forms space of dimension 10, character [zeta3 + 1] and weight 2 over Cyclotomic Field of order 9 and degree 6
        sage: S9.change_ring(CyclotomicField(3))
        Cuspidal subspace of dimension 8 of Modular Forms space of dimension 10, character [zeta3 + 1] and weight 2 over Cyclotomic Field of order 3 and degree 2
        sage: S9.change_ring(QQ)
        Traceback (most recent call last):
        ...
        ValueError: Space cannot be defined over Rational Field

  `is_cuspidal`()

    Return True since spaces of cusp forms are cuspidal.

    EXAMPLES:

        sage: CuspForms(4,10).is_cuspidal()
        True

  `modular_symbols`(sign=0)

    Return the corresponding space of modular symbols with the given sign.

    EXAMPLES:

        sage: S = ModularForms(11,2).cuspidal_submodule()
        sage: S.modular_symbols()
        Modular Symbols subspace of dimension 2 of Modular Symbols space
        of dimension 3 for Gamma_0(11) of weight 2 with sign 0 over Rational Field

        sage: S.modular_symbols(sign=-1)
        Modular Symbols subspace of dimension 1 of Modular Symbols space
        of dimension 1 for Gamma_0(11) of weight 2 with sign -1 over Rational Field

        sage: M = S.modular_symbols(sign=1); M
        Modular Symbols subspace of dimension 1 of Modular Symbols space of
        dimension 2 for Gamma_0(11) of weight 2 with sign 1 over Rational Field
        sage: M.sign()
        1

        sage: S = ModularForms(1,12).cuspidal_submodule()
        sage: S.modular_symbols()
        Modular Symbols subspace of dimension 2 of Modular Symbols space of
        dimension 3 for Gamma_0(1) of weight 12 with sign 0 over Rational Field

        sage: eps = DirichletGroup(13).0
        sage: S = CuspForms(eps^2, 2)

        sage: S.modular_symbols(sign=0)
        Modular Symbols subspace of dimension 2 of Modular Symbols space of dimension 4 and level 13, weight 2, character [zeta6], sign 0, over Cyclotomic Field of order 6 and degree 2

        sage: S.modular_symbols(sign=1)
        Modular Symbols subspace of dimension 1 of Modular Symbols space of dimension 3 and level 13, weight 2, character [zeta6], sign 1, over Cyclotomic Field of order 6 and degree 2

        sage: S.modular_symbols(sign=-1)
        Modular Symbols subspace of dimension 1 of Modular Symbols space of dimension 1 and level 13, weight 2, character [zeta6], sign -1, over Cyclotomic Field of order 6 and degree 2

class `sage.modular.modform.cuspidal_submodule.``CuspidalSubmodule_R`(ambient_space)

  Bases: `sage.modular.modform.cuspidal_submodule.CuspidalSubmodule`

  Cuspidal submodule over a non-minimal base ring.

class `sage.modular.modform.cuspidal_submodule.``CuspidalSubmodule_eps`(ambient_space)

  Bases: `sage.modular.modform.cuspidal_submodule.CuspidalSubmodule_modsym_qexp`

  Space of cusp forms with given Dirichlet character.

  EXAMPLES:

      sage: S = CuspForms(DirichletGroup(5).0,5); S
      Cuspidal subspace of dimension 1 of Modular Forms space of dimension 3, character [zeta4] and weight 5 over Cyclotomic Field of order 4 and degree 2

      sage: S.basis()
      [
      q + (-zeta4 - 1)*q^2 + (6*zeta4 - 6)*q^3 - 14*zeta4*q^4 + (15*zeta4 + 20)*q^5 + O(q^6)
      ]
      sage: f = S.0
      sage: f.qexp()
      q + (-zeta4 - 1)*q^2 + (6*zeta4 - 6)*q^3 - 14*zeta4*q^4 + (15*zeta4 + 20)*q^5 + O(q^6)
      sage: f.qexp(7)
      q + (-zeta4 - 1)*q^2 + (6*zeta4 - 6)*q^3 - 14*zeta4*q^4 + (15*zeta4 + 20)*q^5 + 12*q^6 + O(q^7)
      sage: f.qexp(3)
      q + (-zeta4 - 1)*q^2 + O(q^3)
      sage: f.qexp(2)
      q + O(q^2)
      sage: f.qexp(1)
      O(q^1)

class `sage.modular.modform.cuspidal_submodule.``CuspidalSubmodule_g0_Q`(ambient_space)

  Bases: `sage.modular.modform.cuspidal_submodule.CuspidalSubmodule_modsym_qexp`

  Space of cusp forms for \(\Gamma_0(N)\) over \(\QQ\).

class `sage.modular.modform.cuspidal_submodule.``CuspidalSubmodule_g1_Q`(ambient_space)

  Bases: `sage.modular.modform.cuspidal_submodule.CuspidalSubmodule_gH_Q`

  Space of cusp forms for \(\Gamma_1(N)\) over \(\QQ\).

class `sage.modular.modform.cuspidal_submodule.``CuspidalSubmodule_gH_Q`(ambient_space)

  Bases: `sage.modular.modform.cuspidal_submodule.CuspidalSubmodule_modsym_qexp`

  Space of cusp forms for \(\Gamma_1(N)\) over \(\QQ\).

class `sage.modular.modform.cuspidal_submodule.``CuspidalSubmodule_level1_Q`(ambient_space)

  Bases: `sage.modular.modform.cuspidal_submodule.CuspidalSubmodule`

  Space of cusp forms of level 1 over \(\QQ\).

class `sage.modular.modform.cuspidal_submodule.``CuspidalSubmodule_modsym_qexp`(ambient_space)

  Bases: `sage.modular.modform.cuspidal_submodule.CuspidalSubmodule`

  Cuspidal submodule with q-expansions calculated via modular symbols.

  `new_submodule`(p=None)

    Return the new subspace of this space of cusp forms. This is computed using modular symbols.

    EXAMPLE:

        sage: CuspForms(55).new_submodule()
        Modular Forms subspace of dimension 3 of Modular Forms space of dimension 8 for Congruence Subgroup Gamma0(55) of weight 2 over Rational Field

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