42.8 Submodules of spaces of modular forms

Module: sage.modular.modform.submodule

Submodules of spaces of modular forms

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: M.eisenstein_subspace()
Eisenstein subspace of dimension 11 of Modular Forms space of dimension 13
for Congruence Subgroup Gamma1(13) of weight 2 over Rational Field
sage: M == loads(dumps(M))
True
sage: M.cuspidal_subspace()
Cuspidal subspace of dimension 2 of Modular Forms space of dimension 13 for
Congruence Subgroup Gamma1(13) of weight 2 over Rational Field

Class: ModularFormsSubmodule

class ModularFormsSubmodule
A submodule of an ambient space of modular forms.
ModularFormsSubmodule( self, ambient_module, submodule, [dual=None], [check=False])

ambient_module - ModularFormsSpace submodule - a submodule of the ambient space. dual_module - (default: None) ignored check - (default: False) whether to check that the submodule is Hecke equivariant

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: M.eisenstein_subspace()
Eisenstein subspace of dimension 11 of Modular Forms space of dimension 13
for Congruence Subgroup Gamma1(13) of weight 2 over Rational Field

Functions: change_ring

change_ring( self, base_ring)

Return the base change of this subspace of modular forms to base_ring.

sage: M = ModularForms(6,4) ; M.cuspidal_subspace().change_ring(GF(3))
Traceback (most recent call last):
...
NotImplementedError: Base change only currently implemented for ambient
spaces.

Special Functions: __init__,$ \,$ _compute_coefficients,$ \,$ _compute_q_expansion_basis,$ \,$ _repr_

_compute_coefficients( self, element, X)

Compute all coefficients of the modular form element in self for indices in X.

TODO: Implement this function.

sage: M = ModularForms(6,4).cuspidal_subspace()
sage: M._compute_coefficients( M.basis()[0], range(1,100) )
Traceback (most recent call last):
...
NotImplementedError

_compute_q_expansion_basis( self, prec)

Compute q_expansions to precision prec for each element in self.basis().

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.eisenstein_subspace(); S
Eisenstein subspace of dimension 11 of Modular Forms space of dimension 13
for Congruence Subgroup Gamma1(13) of weight 2 over Rational Field

sage: S._compute_q_expansion_basis(5)
[1 + O(q^5),
q + O(q^5),
q^2 + O(q^5),
q^3 + O(q^5),
q^4 + O(q^5),
O(q^5),
O(q^5),
O(q^5),
O(q^5),
O(q^5),
O(q^5)]

_repr_( self)

sage: ModularForms(Gamma1(13),2).eisenstein_subspace()._repr_()
'Eisenstein subspace of dimension 11 of Modular Forms space of dimension 13
for Congruence Subgroup Gamma1(13) of weight 2 over Rational Field'

Class: ModularFormsSubmoduleWithBasis

class ModularFormsSubmoduleWithBasis
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