Subspace of ambient space of modular symbols
INPUT:
EXAMPLES:
sage: M = ModularSymbols(15,4) ; S = M.cuspidal_submodule() # indirect doctest
sage: S
Modular Symbols subspace of dimension 8 of Modular Symbols space of dimension 12 for Gamma_0(15) of weight 4 with sign 0 over Rational Field
sage: S == loads(dumps(S))
True
sage: M = ModularSymbols(1,24)
sage: A = M.ambient_hecke_module()
sage: B = A.submodule([ x.element() for x in M.cuspidal_submodule().gens() ])
sage: S = sage.modular.modsym.subspace.ModularSymbolsSubspace(A, B.free_module())
sage: S
Modular Symbols subspace of dimension 4 of Modular Symbols space of dimension 5 for Gamma_0(1) of weight 24 with sign 0 over Rational Field
sage: S == loads(dumps(S))
True
Return the subspace of self that is fixed under the star involution.
INPUT:
OUTPUT: subspace of modular symbols
EXAMPLES:
sage: S = ModularSymbols(100,2).cuspidal_submodule() ; S
Modular Symbols subspace of dimension 14 of Modular Symbols space of dimension 31 for Gamma_0(100) of weight 2 with sign 0 over Rational Field
sage: S._compute_sign_subspace(1)
Modular Symbols subspace of dimension 7 of Modular Symbols space of dimension 31 for Gamma_0(100) of weight 2 with sign 0 over Rational Field
sage: S._compute_sign_subspace(-1)
Modular Symbols subspace of dimension 7 of Modular Symbols space of dimension 31 for Gamma_0(100) of weight 2 with sign 0 over Rational Field
sage: S._compute_sign_subspace(-1).sign()
-1
Return the string representation of self.
EXAMPLES:
sage: ModularSymbols(24,4).cuspidal_subspace()._repr_()
'Modular Symbols subspace of dimension 16 of Modular Symbols space of dimension 24 for Gamma_0(24) of weight 4 with sign 0 over Rational Field'
Used internally to declare that a given submodule is cuspidal.
EXAMPLES: We abuse this command:
sage: M = ModularSymbols(389)
sage: S = M.eisenstein_submodule()
sage: S._set_is_cuspidal(True)
sage: S.is_cuspidal()
True
The boundary map to the corresponding space of boundary modular symbols. (This is the restriction of the map on the ambient space.)
EXAMPLES:
sage: M = ModularSymbols(1, 24, sign=1) ; M
Modular Symbols space of dimension 3 for Gamma_0(1) of weight 24 with sign 1 over Rational Field
sage: M.basis()
([X^18*Y^4,(0,0)], [X^20*Y^2,(0,0)], [X^22,(0,0)])
sage: M.cuspidal_submodule().basis()
([X^18*Y^4,(0,0)], [X^20*Y^2,(0,0)])
sage: M.eisenstein_submodule().basis()
([X^18*Y^4,(0,0)] + 166747/324330*[X^20*Y^2,(0,0)] + 236364091/6742820700*[X^22,(0,0)],)
sage: M.boundary_map()
Hecke module morphism boundary map defined by the matrix
[ 0]
[ 0]
[-1]
Domain: Modular Symbols space of dimension 3 for Gamma_0(1) of weight ...
Codomain: Space of Boundary Modular Symbols for Modular Group SL(2,Z) ...
sage: M.cuspidal_subspace().boundary_map()
Hecke module morphism defined by the matrix
[0]
[0]
Domain: Modular Symbols subspace of dimension 2 of Modular Symbols space ...
Codomain: Space of Boundary Modular Symbols for Modular Group SL(2,Z) ...
sage: M.eisenstein_submodule().boundary_map()
Hecke module morphism defined by the matrix
[-236364091/6742820700]
Domain: Modular Symbols subspace of dimension 1 of Modular Symbols space ...
Codomain: Space of Boundary Modular Symbols for Modular Group SL(2,Z) ...
Return the cuspidal subspace of this subspace of modular symbols.
EXAMPLES:
sage: S = ModularSymbols(42,4).cuspidal_submodule() ; S
Modular Symbols subspace of dimension 40 of Modular Symbols space of dimension 48 for Gamma_0(42) of weight 4 with sign 0 over Rational Field
sage: S.is_cuspidal()
True
sage: S.cuspidal_submodule()
Modular Symbols subspace of dimension 40 of Modular Symbols space of dimension 48 for Gamma_0(42) of weight 4 with sign 0 over Rational Field
The cuspidal submodule of the cuspidal submodule is just itself:
sage: S.cuspidal_submodule() is S
True
sage: S.cuspidal_submodule() == S
True
An example where we abuse the _set_is_cuspidal function:
sage: M = ModularSymbols(389)
sage: S = M.eisenstein_submodule()
sage: S._set_is_cuspidal(True)
sage: S.cuspidal_submodule()
Modular Symbols subspace of dimension 1 of Modular Symbols space of dimension 65 for Gamma_0(389) of weight 2 with sign 0 over Rational Field
Return the matrix of the dual star involution, which is induced by complex conjugation on the linear dual of modular symbols.
EXAMPLES:
sage: S = ModularSymbols(6,4) ; S.dual_star_involution_matrix()
[ 1 0 0 0 0 0]
[ 0 1 0 0 0 0]
[ 0 -2 1 2 0 0]
[ 0 2 0 -1 0 0]
[ 0 -2 0 2 1 0]
[ 0 2 0 -2 0 1]
sage: S.star_involution().matrix().transpose() == S.dual_star_involution_matrix()
True
Return the Eisenstein subspace of this space of modular symbols.
EXAMPLES:
sage: ModularSymbols(24,4).eisenstein_subspace()
Modular Symbols subspace of dimension 8 of Modular Symbols space of dimension 24 for Gamma_0(24) of weight 4 with sign 0 over Rational Field
sage: ModularSymbols(20,2).cuspidal_subspace().eisenstein_subspace()
Modular Symbols subspace of dimension 0 of Modular Symbols space of dimension 7 for Gamma_0(20) of weight 2 with sign 0 over Rational Field
Returns a list of pairs where
is simple
spaces of modular symbols and self is isomorphic to the direct sum
of the
as a module over the anemic Hecke algebra
adjoin the star involution.
The cuspidal are all simple, but the Eisenstein factors
need not be simple.
The factors are sorted by dimension - don’t depend on much more for now.
ASSUMPTION: self is a module over the anemic Hecke algebra.
EXAMPLES: Note that if the sign is 1 then the cuspidal factors occur twice, one with each star eigenvalue.
sage: M = ModularSymbols(11)
sage: D = M.factorization(); D
(Modular Symbols subspace of dimension 1 of Modular Symbols space of dimension 3 for Gamma_0(11) of weight 2 with sign 0 over Rational Field) *
(Modular Symbols subspace of dimension 1 of Modular Symbols space of dimension 3 for Gamma_0(11) of weight 2 with sign 0 over Rational Field) *
(Modular Symbols subspace of dimension 1 of Modular Symbols space of dimension 3 for Gamma_0(11) of weight 2 with sign 0 over Rational Field)
sage: [A.T(2).matrix() for A, _ in D]
[[-2], [3], [-2]]
sage: [A.star_eigenvalues() for A, _ in D]
[[-1], [1], [1]]
In this example there is one old factor squared.
sage: M = ModularSymbols(22,sign=1)
sage: S = M.cuspidal_submodule()
sage: S.factorization()
(Modular Symbols subspace of dimension 1 of Modular Symbols space of dimension 2 for Gamma_0(11) of weight 2 with sign 1 over Rational Field)^2
sage: M = ModularSymbols(Gamma0(22), 2, sign=1)
sage: M1 = M.decomposition()[1]
sage: M1.factorization()
Modular Symbols subspace of dimension 3 of Modular Symbols space of dimension 5 for Gamma_0(22) of weight 2 with sign 1 over Rational Field
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
Return True if self is cuspidal.
EXAMPLES:
sage: ModularSymbols(42,4).cuspidal_submodule().is_cuspidal()
True
sage: ModularSymbols(12,6).eisenstein_submodule().is_cuspidal()
False
Return True if self is an Eisenstein subspace.
EXAMPLES:
sage: ModularSymbols(22,6).cuspidal_submodule().is_eisenstein()
False
sage: ModularSymbols(22,6).eisenstein_submodule().is_eisenstein()
True
Return the star involution on self, which is induced by complex conjugation on modular symbols.
EXAMPLES:
sage: M = ModularSymbols(1,24)
sage: M.star_involution()
Hecke module morphism Star involution on Modular Symbols space of dimension 5 for Gamma_0(1) of weight 24 with sign 0 over Rational Field defined by the matrix
[ 1 0 0 0 0]
[ 0 -1 0 0 0]
[ 0 0 1 0 0]
[ 0 0 0 -1 0]
[ 0 0 0 0 1]
Domain: Modular Symbols space of dimension 5 for Gamma_0(1) of weight ...
Codomain: Modular Symbols space of dimension 5 for Gamma_0(1) of weight ...
sage: M.cuspidal_subspace().star_involution()
Hecke module morphism defined by the matrix
[ 1 0 0 0]
[ 0 -1 0 0]
[ 0 0 1 0]
[ 0 0 0 -1]
Domain: Modular Symbols subspace of dimension 4 of Modular Symbols space ...
Codomain: Modular Symbols subspace of dimension 4 of Modular Symbols space ...
sage: M.plus_submodule().star_involution()
Hecke module morphism defined by the matrix
[1 0 0]
[0 1 0]
[0 0 1]
Domain: Modular Symbols subspace of dimension 3 of Modular Symbols space ...
Codomain: Modular Symbols subspace of dimension 3 of Modular Symbols space ...
sage: M.minus_submodule().star_involution()
Hecke module morphism defined by the matrix
[-1 0]
[ 0 -1]
Domain: Modular Symbols subspace of dimension 2 of Modular Symbols space ...
Codomain: Modular Symbols subspace of dimension 2 of Modular Symbols space ...