Submodule of a Hecke module.
Sum of self and other (as submodules of a common ambient module).
EXAMPLES:
sage: M = ModularForms(4,10)
sage: S = sage.modular.hecke.submodule.HeckeSubmodule(M, M.submodule(M.basis()[:3]).free_module())
sage: E = sage.modular.hecke.submodule.HeckeSubmodule(M, M.submodule(M.basis()[3:]).free_module())
sage: S + E # indirect doctest
Modular Forms subspace of dimension 6 of Modular Forms space of dimension 6 for Congruence Subgroup Gamma0(4) of weight 10 over Rational Field
Coerce x into the ambient module and checks that x is in this submodule.
EXAMPLES:
sage: M = ModularSymbols(37)
sage: S = M.cuspidal_submodule()
sage: M([0,oo])
-(1,0)
sage: S([0,oo])
...
TypeError: x does not coerce to an element of this Hecke module
sage: S([-1/23,0])
(1,23)
Compare self to other. Returns 0 if self is the same as other, and -1 otherwise.
Initialise a submodule of an ambient Hecke module.
INPUT:
EXAMPLES:
sage: CuspForms(1,60) # indirect doctest
Cuspidal subspace of dimension 5 of Modular Forms space of dimension 6 for Modular Group SL(2,Z) of weight 60 over Rational Field
sage: M = ModularForms(4,10)
sage: S = sage.modular.hecke.submodule.HeckeSubmodule(M, M.submodule(M.basis()[:3]).free_module())
sage: S
Rank 3 submodule of a Hecke module of level 4
sage: S == loads(dumps(S))
True
Compute the Atkin-Lehner matrix corresponding to the divisor d of the level of self.
EXAMPLES:
sage: M = ModularSymbols(4,10)
sage: S = sage.modular.hecke.submodule.HeckeSubmodule(M, M.cuspidal_submodule().free_module())
sage: S
Rank 6 submodule of a Hecke module of level 4
sage: S._compute_atkin_lehner_matrix(1)
[1 0 0 0 0 0]
[0 1 0 0 0 0]
[0 0 1 0 0 0]
[0 0 0 1 0 0]
[0 0 0 0 1 0]
[0 0 0 0 0 1]
Compute the matrix for the nth Hecke operator acting on the dual of self.
EXAMPLES:
sage: M = ModularForms(4,10)
sage: S = sage.modular.hecke.submodule.HeckeSubmodule(M, M.submodule(M.basis()[:3]).free_module())
sage: S._compute_dual_hecke_matrix(3)
[ 0 0 1]
[ 0 -156 0]
[35568 0 72]
sage: CuspForms(4,10).dual_hecke_matrix(3)
[ 0 0 1]
[ 0 -156 0]
[35568 0 72]
Compute the matrix of the nth Hecke operator acting on this space, by calling the corresponding function for the ambient space and restricting. If n is not coprime to the level, we check that the restriction is well-defined.
EXAMPLES:
sage: R.<q> = QQ[[]]
sage: M = ModularForms(2, 12)
sage: f = M(q^2 - 24*q^4 + O(q^6))
sage: A = M.submodule(M.free_module().span([f.element()]),check=False)
sage: sage.modular.hecke.submodule.HeckeSubmodule._compute_hecke_matrix(A, 3)
[252]
sage: sage.modular.hecke.submodule.HeckeSubmodule._compute_hecke_matrix(A, 4)
...
ArithmeticError: subspace is not invariant under matrix
String representation of self.
EXAMPLES:
sage: M = ModularForms(4,10)
sage: S = sage.modular.hecke.submodule.HeckeSubmodule(M, M.submodule(M.basis()[:3]).free_module())
sage: S._repr_()
'Rank 3 submodule of a Hecke module of level 4'
Set the dual free module of self to V. Here V must be a vector space of the same dimension as self, embedded in a space of the same dimension as the ambient space of self.
EXAMPLES:
sage: M = ModularSymbols(4,10)
sage: S = sage.modular.hecke.submodule.HeckeSubmodule(M, M.cuspidal_submodule().free_module())
sage: S._set_dual_free_module(M.cuspidal_submodule().dual_free_module())
sage: S._set_dual_free_module(S)
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
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
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
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:
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
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)
...
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
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]
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
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
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
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
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
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)
Alias for code{self.free_module()}.
EXAMPLES:
sage: M = ModularSymbols(17,4).cuspidal_subspace()
sage: M.free_module() is M.module()
True
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
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
Return the old or p-old submodule of this space of modular symbols.
EXAMPLES: We compute the old and new submodules of
.
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
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
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
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:
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
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