The Hecke Algebra

HeckeBound(M) : ModSym -> RngIntElt

A positive integer n such that the Hecke operators T_1, ..., T_n generate the Hecke algebra as a Z-module. When the character is trivial, the default bound is (k/12)⋅[( SL)_2(Z):Gamma_0(N)]. That this suffices follows from [Stu87], as is explained in [AS]. When the character of M is nontrivial, the default bound is twice the above bound; however, it is not known that this bound is large enough in all cases in which the character is nontrivial, so one may wish to increase the bound using SetHeckeBound.

SetHeckeBound(M, n) : ModSym, RngIntElt -> RngIntElt

Many computations require a bound n such that T_1, ..., T_n generate the Hecke algebra as a Z-module. This command allows you to set the bound that is used internally. Setting it too low can result in functions quickly producing incorrect results.

HeckeAlgebra(M : Bound) : ModSym -> AlgMat

The Hecke algebra associated to M. This is an algebra TQ over Q, such that Generators(TQ) is a set that generates the ring Z[T_1, T_2, T_3, ... ], as a Z-module. If the optional integer parameter Bound is set, then HeckeAlgebra only computes the algebra generated by those T_n, with n <= Bound.

DiscriminantOfHeckeAlgebra(M : Bound) : ModSym -> RngIntElt

The discriminant of the Hecke algebra associated to M. If the optional parameter Bound is set, then the discriminant of the algebra generated by only those T_n, with n <= Bound, is computed instead.

HeckeEigenvalueRing(M : parameters) : ModSym -> Rng, Map

    Bound: RngIntElt                    Default: -1

The order generated by the Fourier coefficients of one of the q-expansions of a newform corresponding to M, along with a map from the ring containing the coefficients of qExpansion(A) to the order. If the optional parameter Bound is set, then the order generated only by those a_n, with n <= Bound, is computed.

HeckeEigenvalueField(M) : ModSym -> Fld, Map

The number field generated by the Fourier coefficients of one of the q-expansions of a newform corresponding to M, along with a map from the ring containing the coefficients of qExpansion(M) to the number field. We require that M be defined over Q.

Example ModSym_HeckeAlgebra (H88E17)

> M := ModularSymbols(389,2,+1);
> C := CuspidalSubspace(M);
> DiscriminantOfHeckeAlgebra(C);
62967005472006188288017473632139259549820493155023510831104000000
> Factorization($1);
[ <2, 53>, <3, 4>, <5, 6>, <31, 2>, <37, 1>, <389, 1>, <3881, 1>, 
<215517113148241, 1>, <477439237737571441, 1> ]