Operators

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.

HeckeOperator(M, n) : ModFrm, RngIntElt -> AlgMatElt
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.)
HeckeOperator(n,f) : RngIntElt, ModFrmElt -> ModFrmElt
The image under the Hecke operator Tn of the given modular form.
HeckePolynomial(M, n : parameters) : ModFrm, RngIntElt -> RngUPolElt
    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.
AtkinLehnerOperator(M, q) : ModFrm, RngIntElt -> AlgMatElt
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.)
AtkinLehnerOperator(q,f) : RngIntElt, ModFrmElt -> ModFrmElt
The image under the involution wq of the given modular form.

Example ModFrm_HeckePolynomials (H144E13)

First we compute a characteristic polynomial on S2(Γ1(13)) over both Z and the finite field F2. 

> 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
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Version: V2.29 of Fri Nov 28 15:14:01 AEDT 2025
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