Each space M of modular forms comes equipped with a commuting family T1, T2, T3, ... of linear operators acting on it called the Hecke operators. Unfortunately, at present, the computation of Hecke and other operators on spaces of modular forms with nontrivial character has not yet been implemented, though computation of characteristic polynomials of Hecke operators is supported.
The matrix representing the nth Hecke operator Tn with respect to Basis(M). (Currently M must be a space of modular forms with trivial character and integral weight ≥2.)
The image under the Hecke operator Tn of the given modular form.
Proof: BoolElt Default: true
The characteristic polynomial of the nth Hecke operator Tn. In some situations this is more efficient than CharacteristicPolynomial(HeckeOperator(M,n)) or any of its variants. Note that M can be an arbitrary space of modular forms.
The matrix representing the qth Atkin-Lehner involution Wq on M with respect to Basis(M). (Currently M must be a cuspidal space of modular forms with trivial character and integral weight ≥2.)
The image under the involution wq of the given modular form.
> R<x> := PolynomialRing(Integers());
> S := CuspForms(Gamma1(13),2);
> HeckePolynomial(S, 2);
x^2 + 3*x + 3
> S2 := BaseExtend(S, GF(2));
> R<y> := PolynomialRing(GF(2));
> Factorization(HeckePolynomial(S2,2));
[
<y^2 + y + 1, 1>
]
Next we compute a Hecke operator on M4(Γ0(14)).
> M := ModularForms(Gamma0(14),4);
> T := HeckeOperator(M,2);
> T;
[ 1 0 0 0 0 0 0 240]
[ 0 0 0 0 18 12 50 100]
[ 0 1 0 0 -2 18 12 -11]
[ 0 0 0 0 1 22 25 46]
[ 0 0 1 0 -1 -16 -20 -82]
[ 0 0 0 0 -1 -6 -9 -38]
[ 0 0 0 1 3 9 15 39]
[ 0 0 0 0 0 0 0 8]
> Parent(T);
Full Matrix Algebra of degree 8 over Integer Ring
> Factorization(CharacteristicPolynomial(T));
[
<x - 8, 2>,
<x - 2, 1>,
<x - 1, 2>,
<x + 2, 1>,
<x^2 + x + 8, 1>
]
> f := M.1;
> f*T;
1 + 240*q^7 + O(q^8)
> M.1 + 240*M.8;
1 + 240*q^7 + O(q^8)
This example demonstrates the Atkin-Lehner involution W3 on S2(Γ0(33)).
> M := ModularForms(33,2);
> S := CuspidalSubspace(M);
> W3 := AtkinLehnerOperator(S, 3);
> W3;
[ 1 0 0]
[ 1/3 1/3 -4/3]
[ 1/3 -2/3 -1/3]
> Factorization(CharacteristicPolynomial(W3));
[
<x - 1, 2>,
<x + 1, 1>
]
> f := S.2;
> f*W3;
1/3*q + 1/3*q^2 - 4/3*q^3 - 1/3*q^4 - 2/3*q^5 + 5/3*q^6
+ 4/3*q^7 + O(q^8)
The Atkin-Lehner and Hecke operators need not commute:
> T3 := HeckeOperator(S, 3); > T3; [ 0 -2 -1] [ 0 -1 1] [ 1 -2 -1] > T3*W3 - W3*T3 eq 0; false