# Submodules of Hecke modules¶

class sage.modular.hecke.submodule.HeckeSubmodule(ambient, submodule, dual_free_module=None, check=True)

Submodule of a Hecke module.

ambient()

Synonym for ambient_hecke_module.

EXAMPLES:

sage: CuspForms(2, 12).ambient()
Modular Forms space of dimension 4 for Congruence Subgroup Gamma0(2) of weight 12 over Rational Field

ambient_hecke_module()

Return the ambient Hecke module of which this is a submodule.

EXAMPLES:

sage: CuspForms(2, 12).ambient_hecke_module()
Modular Forms space of dimension 4 for Congruence Subgroup Gamma0(2) of weight 12 over Rational Field

complement(bound=None)

Return the largest Hecke-stable complement of this space.

EXAMPLES:

sage: M = ModularSymbols(15, 6).cuspidal_subspace()
sage: M.complement()
Modular Symbols subspace of dimension 4 of Modular Symbols space of dimension 20 for Gamma_0(15) of weight 6 with sign 0 over Rational Field
sage: E = EllipticCurve("128a")
sage: ME = E.modular_symbol_space()
sage: ME.complement()
Modular Symbols subspace of dimension 17 of Modular Symbols space of dimension 18 for Gamma_0(128) of weight 2 with sign 1 over Rational Field

degeneracy_map(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.

EXAMPLES:

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(bound=None, anemic=True, use_star=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.

If use_star is True (which is the default), we also use the +/- eigenspaces for the star operator to find the dual free module of self. If self does not have a star involution, use_star will automatically be set to False.

EXAMPLES:

sage: M = ModularSymbols(11, 2)
sage: M.dual_free_module()
Vector space of dimension 3 over Rational Field
sage: Mpc = M.plus_submodule().cuspidal_submodule()
sage: Mcp = M.cuspidal_submodule().plus_submodule()
sage: Mcp.dual_free_module() == Mpc.dual_free_module()
True
sage: Mpc.dual_free_module()
Vector space of degree 3 and dimension 1 over Rational Field
Basis matrix:
[  1 5/2   5]

sage: M = ModularSymbols(35,2).cuspidal_submodule()
sage: M.dual_free_module(use_star=False)
Vector space of degree 9 and dimension 6 over Rational Field
Basis matrix:
[   1    0    0    0   -1    0    0    4   -2]
[   0    1    0    0    0    0    0 -1/2  1/2]
[   0    0    1    0    0    0    0 -1/2  1/2]
[   0    0    0    1   -1    0    0    1    0]
[   0    0    0    0    0    1    0   -2    1]
[   0    0    0    0    0    0    1   -2    1]

sage: M = ModularSymbols(40,2)
sage: Mmc = M.minus_submodule().cuspidal_submodule()
sage: Mcm = M.cuspidal_submodule().minus_submodule()
sage: Mcm.dual_free_module() == Mmc.dual_free_module()
True
sage: Mcm.dual_free_module()
Vector space of degree 13 and dimension 3 over Rational Field
Basis matrix:
[ 0  1  0  0  0  0  1  0 -1 -1  1 -1  0]
[ 0  0  1  0 -1  0 -1  0  1  0  0  0  0]
[ 0  0  0  0  0  1  1  0 -1  0  0  0  0]

sage: M = ModularSymbols(43).cuspidal_submodule()
sage: S = M[0].plus_submodule() + M[1].minus_submodule()
sage: S.dual_free_module(use_star=False)
Traceback (most recent call last):
...
RuntimeError: Computation of complementary space failed (cut down to rank 7, but should have cut down to rank 4).
sage: S.dual_free_module().dimension() == S.dimension()
True


We test that #5080 is fixed:

sage: EllipticCurve('128a').congruence_number()
32

free_module()

Return the free module corresponding to self.

EXAMPLES:

sage: M = ModularSymbols(33,2).cuspidal_subspace() ; M
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: M.free_module()
Vector space of degree 9 and dimension 6 over Rational Field
Basis matrix:
[ 0  1  0  0  0  0  0 -1  1]
[ 0  0  1  0  0  0  0 -1  1]
[ 0  0  0  1  0  0  0 -1  1]
[ 0  0  0  0  1  0  0 -1  1]
[ 0  0  0  0  0  1  0 -1  1]
[ 0  0  0  0  0  0  1 -1  0]

hecke_bound()

Compute the Hecke bound for self; that is, a number n such that the T_m for m = n generate the Hecke algebra.

EXAMPLES:

sage: M = ModularSymbols(24,8)
sage: M.hecke_bound()
53
sage: M.cuspidal_submodule().hecke_bound()
32
sage: M.eisenstein_submodule().hecke_bound()
53

intersection(other)

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

EXAMPLES:

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_ambient()

Return True if self is an ambient space of modular symbols.

EXAMPLES:

sage: M = ModularSymbols(17,4)
sage: M.cuspidal_subspace().is_ambient()
False
sage: A = M.ambient_hecke_module()
sage: S = A.submodule(A.basis())
sage: sage.modular.hecke.submodule.HeckeSubmodule.is_ambient(S)
True

is_new(p=None)

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

EXAMPLES:

sage: M = ModularSymbols(1,16)
sage: S = sage.modular.hecke.submodule.HeckeSubmodule(M, M.cuspidal_submodule().free_module())
sage: S.is_new()
True

is_old(p=None)

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

EXAMPLES:

sage: M = ModularSymbols(50,2)
sage: S = sage.modular.hecke.submodule.HeckeSubmodule(M, M.old_submodule().free_module())
sage: S.is_old()
True
sage: S = sage.modular.hecke.submodule.HeckeSubmodule(M, M.new_submodule().free_module())
sage: S.is_old()
False

is_submodule(V)

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

EXAMPLES:

sage: M = ModularSymbols(30,4)
sage: S = sage.modular.hecke.submodule.HeckeSubmodule(M, M.cuspidal_submodule().free_module())
sage: S.is_submodule(M)
True
sage: SS = sage.modular.hecke.submodule.HeckeSubmodule(M, M.old_submodule().free_module())
sage: S.is_submodule(SS)
False

linear_combination_of_basis(v)

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

EXAMPLES:

sage: M = ModularForms(Gamma0(2),12)
sage: S = sage.modular.hecke.submodule.HeckeSubmodule(M, M.cuspidal_submodule().free_module())
sage: S.basis()
(q + 252*q^3 - 2048*q^4 + 4830*q^5 + O(q^6), q^2 - 24*q^4 + O(q^6))
sage: S.linear_combination_of_basis([3,10])
3*q + 10*q^2 + 756*q^3 - 6384*q^4 + 14490*q^5 + O(q^6)

module()

Alias for code{self.free_module()}.

EXAMPLES:

sage: M = ModularSymbols(17,4).cuspidal_subspace()
sage: M.free_module() is M.module()
True

new_submodule(p=None)

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

EXAMPLES:

sage: M = ModularSymbols(20,4)
sage: M.new_submodule()
Modular Symbols subspace of dimension 2 of Modular Symbols space of dimension 18 for Gamma_0(20) of weight 4 with sign 0 over Rational Field
sage: S = sage.modular.hecke.submodule.HeckeSubmodule(M, M.cuspidal_submodule().free_module())
sage: S
Rank 12 submodule of a Hecke module of level 20
sage: S.new_submodule()
Modular Symbols subspace of dimension 2 of Modular Symbols space of dimension 18 for Gamma_0(20) of weight 4 with sign 0 over Rational Field

nonembedded_free_module()

Return the free module corresponding to self as an abstract free module, i.e. not as an embedded vector space.

EXAMPLES:

sage: M = ModularSymbols(12,6)
sage: S = sage.modular.hecke.submodule.HeckeSubmodule(M, M.cuspidal_submodule().free_module())
sage: S
Rank 14 submodule of a Hecke module of level 12
sage: S.nonembedded_free_module()
Vector space of dimension 14 over Rational Field

old_submodule(p=None)

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

EXAMPLES: 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

rank()

Return the rank of self as a free module over the base ring.

EXAMPLE:

sage: ModularSymbols(6, 4).cuspidal_subspace().rank()
2
sage: ModularSymbols(6, 4).cuspidal_subspace().dimension()
2

submodule(M, Mdual=None, check=True)

Construct a submodule of self from the free module M, which must be a subspace of self.

EXAMPLES:

sage: M = ModularSymbols(18,4)
sage: S = sage.modular.hecke.submodule.HeckeSubmodule(M, M.cuspidal_submodule().free_module())
sage: S[0]
Modular Symbols subspace of dimension 2 of Modular Symbols space of dimension 18 for Gamma_0(18) of weight 4 with sign 0 over Rational Field
sage: S.submodule(S[0].free_module())
Modular Symbols subspace of dimension 2 of Modular Symbols space of dimension 18 for Gamma_0(18) of weight 4 with sign 0 over Rational Field

submodule_from_nonembedded_module(V, Vdual=None, check=True)

Construct a submodule of self from V. Here V should be a subspace of a vector space whose dimension is the same as that of self.

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

EXAMPLES:

sage: M = ModularSymbols(37,2)
sage: S = sage.modular.hecke.submodule.HeckeSubmodule(M, M.cuspidal_submodule().free_module())
sage: V = (QQ**4).subspace([[1,-1,0,1/2],[0,0,1,-1/2]])
sage: S.submodule_from_nonembedded_module(V)
Modular Symbols subspace of dimension 2 of Modular Symbols space of dimension 5 for Gamma_0(37) of weight 2 with sign 0 over Rational Field

sage.modular.hecke.submodule.is_HeckeSubmodule(x)

Return True if x is of type HeckeSubmodule.

EXAMPLES:

sage: sage.modular.hecke.submodule.is_HeckeSubmodule(ModularForms(1, 12))
False
sage: sage.modular.hecke.submodule.is_HeckeSubmodule(CuspForms(1, 12))
True


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