40.7 Submodule of a Hecke module

Module: sage.modular.hecke.submodule

Submodule of a Hecke module.

Module-level Functions

is_HeckeSubmodule( x)

Class: HeckeSubmodule

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

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.

sage: D = ModularSymbols(10,4).cuspidal_submodule().decomposition(); D
[
Modular Symbols subspace of dimension 2 of Modular Symbols space of
dimension 10 for Gamma_0(10) of weight 4 with sign 0 over Rational Field,
Modular Symbols subspace of dimension 4 of Modular Symbols space of
dimension 10 for Gamma_0(10) of weight 4 with sign 0 over Rational Field
]
sage: d = D[1].degeneracy_map(5); d
Hecke module morphism defined by the matrix
[   0    0   -1    1]
[   0  1/2  3/2   -2]
[   0   -1    1    0]
[   0 -3/4 -1/4    1]
Domain: Modular Symbols subspace of dimension 4 of Modular Symbols space
...
Codomain: Modular Symbols space of dimension 4 for Gamma_0(5) of weight ...

sage: d.rank()
2
sage: d.kernel()
Modular Symbols subspace of dimension 2 of Modular Symbols space of
dimension 10 for Gamma_0(10) of weight 4 with sign 0 over Rational Field
sage: d.image()
Modular Symbols subspace of dimension 2 of Modular Symbols space of
dimension 4 for Gamma_0(5) of weight 4 with sign 0 over Rational Field

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

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

TESTS:

sage: M = ModularSymbols(1,80)
sage: M.plus_submodule().cuspidal_submodule().sign() # indirect doctest
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=None], [check=True])

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

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

Input:

V
- submodule of ambient free module of the same rank as the rank of self.
check
- whether to check that V is Hecke equivariant.

Output: Hecke submodule of self

Special Functions: __add__,$ \,$ __call__,$ \,$ __cmp__,$ \,$ __init__,$ \,$ _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=True])

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

sage: M = ModularSymbols(37)
sage: S = M.cuspidal_submodule()
sage: M([0,oo])
-(1,0)
sage: S([0,oo])
Traceback (most recent call last):
...
TypeError: x does not coerce to an element of this Hecke module
sage: S([-1/23,0])
(1,23)

_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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