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 \(\Gamma_H(N)\) 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.

hecke_polynomial(n, var='x')¶

Return the characteristic polynomial of the Hecke operator T_n on this space. This is computed via modular symbols, and in particular is faster to compute than the matrix itself.

EXAMPLES:

sage: CuspForms(105, 2).hecke_polynomial(2, 'y')
y^13 + 5*y^12 - 4*y^11 - 52*y^10 - 34*y^9 + 174*y^8 + 212*y^7 - 196*y^6 - 375*y^5 - 11*y^4 + 200*y^3 + 80*y^2

Check that this gives the same answer as computing the Hecke matrix:

sage: CuspForms(105, 2).hecke_matrix(2).charpoly(var='y')
y^13 + 5*y^12 - 4*y^11 - 52*y^10 - 34*y^9 + 174*y^8 + 212*y^7 - 196*y^6 - 375*y^5 - 11*y^4 + 200*y^3 + 80*y^2

Check that trac ticket #21546 is fixed (this example used to take about 5 hours):

sage: CuspForms(1728, 2).hecke_polynomial(2) # long time (20 sec)
x^253 + x^251 - 2*x^249

new_submodule(p=None)¶

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

EXAMPLES:

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

class sage.modular.modform.cuspidal_submodule.CuspidalSubmodule_wt1_eps(ambient_space)¶

Bases: sage.modular.modform.cuspidal_submodule.CuspidalSubmodule

Space of cusp forms of weight 1 with specified character.

class sage.modular.modform.cuspidal_submodule.CuspidalSubmodule_wt1_gH(ambient_space)¶

Bases: sage.modular.modform.cuspidal_submodule.CuspidalSubmodule

Space of cusp forms of weight 1 for a GammaH group.

The Cuspidal Subspace¶

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