parent) |
apply_sparse,
charpoly,
codomain,
decomposition,
det,
domain,
fcp,
hecke_module_morphism,
image,
kernel,
trace
Further documentation:
x) |
Apply this Hecke operator to x, where we avoid computing the matrix of x if possible.
sage: M = ModularSymbols(11) sage: T = M.hecke_operator(23) sage: T.apply_sparse(M.gen(0)) 24*(1,0) - 5*(1,9)
) |
Return the characteristic polynomial of this Hecke operator.
sage: M = ModularSymbols(Gamma1(6),4) sage: M.hecke_operator(2).charpoly() x^6 - 14*x^5 + 29*x^4 + 172*x^3 - 124*x^2 - 320*x + 256
) |
Decompose the Hecke module under the action of this Hecke operator.
sage: M = ModularSymbols(11) sage: t2 = M.hecke_operator(2) sage: t2.decomposition() [Dimension 1 subspace of a modular symbols space of level 11, Dimension 2 subspace of a modular symbols space of level 11]
) |
Return the determinant of this Hecke operator.
sage: M = ModularSymbols(23) sage: T = M.hecke_operator(3) sage: T.det() 100
) |
Return the factorization of the characteristic polynomial of this Hecke operator.
sage: M = ModularSymbols(23) sage: T = M.hecke_operator(3) sage: T.fcp() (x - 4) * (x^2 - 5)^2
) |
Return the endomorphism of Hecke modules defined by the matrix attached to this Hecke operator.
sage: M = ModularSymbols(Gamma1(13)) sage: t = M.hecke_operator(2) sage: t Hecke operator T_2 on Full Modular Symbols space for Gamma_1(13) of weight 2 with sign 0 and dimension 15 over Rational Field sage: t.hecke_module_morphism() Hecke module morphism T_2 defined by the matrix (not printing 15 x 15 matrix) Domain: Full Modular Symbols space for Gamma_1(13) of weight 2 with sign ... Codomain: Full Modular Symbols space for Gamma_1(13) of weight 2 with sign ...
) |
Return the image of this Hecke operator.
sage: M = ModularSymbols(23) sage: T = M.hecke_operator(3) sage: T.fcp() (x - 4) * (x^2 - 5)^2 sage: T.image() Dimension 5 subspace of a modular symbols space of level 23 sage: (T-4).image() Dimension 4 subspace of a modular symbols space of level 23 sage: (T**2-5).image() Dimension 1 subspace of a modular symbols space of level 23
) |
Return the kernel of this Hecke operator.
sage: M = ModularSymbols(23) sage: T = M.hecke_operator(3) sage: T.fcp() (x - 4) * (x^2 - 5)^2 sage: T.kernel() Dimension 0 subspace of a modular symbols space of level 23 sage: (T-4).kernel() Dimension 1 subspace of a modular symbols space of level 23 sage: (T**2-5).kernel() Dimension 4 subspace of a modular symbols space of level 23
) |
Return the trace of this Hecke operator.
sage: M = ModularSymbols(1,12) sage: T = M.hecke_operator(2) sage: T.trace() 2001
Instances of class HeckeAlgebraElement also have the following special methods:
__add__,
__call__,
__rmul__,
__sub__,
_HeckeAlgebraElement__is_compatible
Further documentation:
other) |
sage: M = ModularSymbols(11) sage: t = M.hecke_operator(2) sage: t Hecke operator T_2 on Full Modular Symbols space for Gamma_0(11) of weight 2 with sign 0 and dimension 3 over Rational Field sage: t + t Hecke operator on Full Modular Symbols space for Gamma_0(11) of weight 2 with sign 0 and dimension 3 over Rational Field defined by: [ 6 0 -2] [ 0 -4 0] [ 0 0 -4]
We can also add Hecke operators with different indexes:
sage: M = ModularSymbols(Gamma1(6),4) sage: t2 = M.hecke_operator(2); t3 = M.hecke_operator(3) sage: t2 - t3 Hecke operator on Full Modular Symbols space for Gamma_1(6) of weight 4 with sign 0 and dimension 6 over Rational Field defined by: (not printing 6 x 6 matrix) sage: (t2 - t3).charpoly() x^6 + 36*x^5 + 104*x^4 - 3778*x^3 + 7095*x^2 - 3458*x
x) |
Apply this Hecke operator to
.
sage: M = ModularSymbols(11); t2 = M.hecke_operator(2) sage: t2(M.gen(0)) 3*(1,0) - (1,9)
sage: t2 = M.hecke_operator(2); t3 = M.hecke_operator(3) sage: t3(t2(M.gen(0))) 12*(1,0) - 2*(1,9) sage: (t3*t2)(M.gen(0)) 12*(1,0) - 2*(1,9)
left) |
sage: M = ModularSymbols(11); t2 = M.hecke_operator(2) sage: 2*t2 Hecke operator on Full Modular Symbols space for Gamma_0(11) of weight 2 with sign 0 and dimension 3 over Rational Field defined by: [ 6 0 -2] [ 0 -4 0] [ 0 0 -4]
other) |
Compute the difference of self and other.
sage: M = ModularSymbols(Gamma1(6),4) sage: t2 = M.hecke_operator(2); t3 = M.hecke_operator(3) sage: t2 - t3 Hecke operator on Full Modular Symbols space for Gamma_1(6) of weight 4 with sign 0 and dimension 6 over Rational Field defined by: (not printing 6 x 6 matrix)
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