The Hecke Algebra

HeckeBound(M): ModSym -> RngIntElt

A positive integer \(n\) such that the Hecke operators \(T_1, \ldots, T_n\) generate the Hecke algebra as a \({\mathbb{Z}}\)-module. When the character is trivial, the default bound is \((k/12)\cdot[{\rm SL}_2({\mathbb{Z}}):\Gamma_0(N)]\). That this suffices follows from [Sturm, 1987], as is explained in [Agashe and Stein, 2002]. When the character of the space of modular symbols \(M\) 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 \(T_1,\ldots, T_n\) generate the Hecke algebra associated to the space of modular symbols \(M\) as a \({\mathbb{Z}}\)-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 the space of modular symbols \(M\). This is an algebra TQ over \({\mathbb{Q}}\), such that Generators(TQ) is a set that generates the ring \({\mathbb{Z}}[T_1,T_2,T_3,\ldots]\), as a \({\mathbb{Z}}\)-module. If the optional integer parameter Bound is set, then HeckeAlgebra only computes the algebra generated by those \(T_n\), with \(n\leq\) Bound.

DiscriminantOfHeckeAlgebra(M : Bound): ModSym -> RngIntElt

The discriminant of the Hecke algebra associated to the space of modular symbols \(M\). If the optional parameter Bound is set, then the discriminant of the algebra generated by only those \(T_n\), with \(n\leq\) 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 the space of modular symbols \(M\), 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 \(n \leq\) 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 the space of modular symbols \(M\), along with a map from the ring containing the coefficients of qExpansion(M) to the number field. We require that \(M\) be defined over \({\mathbb{Q}}\).

Example: Hecke Algebra (ex-1119e4)

In this example, we compute the discriminant of the Hecke algebra of prime level \(389\).

> 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> ]

Run in calculatorCopy input

The prime \(389\) is the only prime \(p<10000\) such that \(p\) divides the discriminant of the Hecke algebra associated to \(S_2(\Gamma_0(p))\). It is an open problem to decide whether or not there are any other such primes. Are there infinitely many?