26.2 Submodule of a Hecke module

Module: sage.modular.hecke.submodule

Module-level Functions

is_HeckeSubmodule( x)

Class: HeckeSubmodule

class HeckeSubmodule
Submodule of a Hecke module.
HeckeSubmodule( self, ambient, submodule, [dual_free_module=True], [check=None])

Functions: ambient,$  $ ambient_hecke_module,$  $ complement,$  $ degeneracy_map,$  $ dimension,$  $ dual_free_module,$  $ free_module,$  $ intersection,$  $ is_ambient,$  $ is_new,$  $ is_old,$  $ is_submodule,$  $ linear_combination_of_basis,$  $ module,$  $ new_submodule,$  $ nonembedded_free_module,$  $ old_submodule,$  $ rank,$  $ submodule,$  $ submodule_from_nonembedded_module

complement( self, [bound=None])

Return the largest Hecke-stable complement of this space.

degeneracy_map( self, level, [t=1])

The t-th degeneracy map from self to the space of ambient modular symbols of the given level. The level of self must be a divisor or multiple of level, and t must be a divisor of the quotient.

INPUT:
    level -- int, the level of the codomain of the map (positive int).
    t  -- int, the parameter of the degeneracy map, i.e., the map is
          related to f(q) |--> f(q^t).
          
OUTPUT:
    A linear function from self to the space of modular symbols
    of given level with the same weight, character, sign,
    etc., as this space.

dual_free_module( self, [bound=True], [anemic=None])

Compute embedded dual free module if possible. In general this won't be possible, e.g., if this space is not Hecke equivariant, possibly if it is not cuspidal, or if the characteristic is not 0. In all these cases we raise a RuntimeError exception.

intersection( self, other)

Returns the intersection of self and other, which must both lie in a common ambient space of modular symbols.

sage: M = ModularSymbols(43, sign=1)
sage: A = M[0] + M[1]
sage: B = M[1] + M[2]
sage: A.dimension(), B.dimension()
(2, 3)

sage: C = A.intersection(B); C.dimension()
1

is_new( self, [p=None])

Returns True if this Hecke module is p-new. If p is None, returns True if it is new.

is_old( self, [p=None])

Returns True if this Hecke module is p-old. If p is None, returns True if it is old.

is_submodule( self, V)

Returns True if and only if self is a submodule of V.

linear_combination_of_basis( self, v)

Return the linear combination of the basis of self given by the entries of v.

new_submodule( self, [p=None])

Return the new or p-new submodule of this space of modular symbols.

old_submodule( self, [p=None])

Return the old or p-old submodule of this space of modular symbols.

We compute the old and new submodules of $ \mathbf{S}_2(\Gamma_0(33))$ .

sage: M = ModularSymbols(33); S = M.cuspidal_submodule(); S
Modular Symbols subspace of dimension 6 of Modular Symbols space of
dimension 9 for Gamma_0(33) of weight 2 with sign 0 over Rational Field
sage: S.old_submodule()
Modular Symbols subspace of dimension 4 of Modular Symbols space of
dimension 9 for Gamma_0(33) of weight 2 with sign 0 over Rational Field
sage: S.new_submodule()
Modular Symbols subspace of dimension 2 of Modular Symbols space of
dimension 9 for Gamma_0(33) of weight 2 with sign 0 over Rational Field

submodule( self, M, [Mdual=True], [check=None])

Construct a submodule of self from the embedded free module M.

submodule_from_nonembedded_module( self, V, [Vdual=True], [check=None])

INPUT:
    V -- submodule of ambient free module of the same rank as the
         rank of self.
OUTPUT:
    Hecke submodule of self

Special Functions: __add__,$  $ __call__,$  $ __cmp__,$  $ _compute_atkin_lehner_matrix,$  $ _compute_dual_hecke_matrix,$  $ _compute_hecke_matrix,$  $ _repr_,$  $ _set_dual_free_module,$  $ _set_dual_free_module_from_nonembedded_module

__call__( self, x, [check=False])

Coerce x into the ambient module and checks that x is in this submodule.

_set_dual_free_module_from_nonembedded_module( self, V)

INPUT:
    V -- submodule of ambient free module of the same rank as the
         rank of self.
OUTPUT:
    Hecke submodule of self

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