Operators

Each space \(M\) of modular forms comes equipped with a commuting family \(T_1, T_2, T_3, \ldots\) 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 \(n\)th Hecke operator \(T_n\) with respect to Basis(M). (Currently \(M\) must be a space of modular forms with trivial character and integral weight \(\geq 2\).)

HeckeOperator(n, f): RngIntElt, ModFrmElt -> ModFrmElt

The image under the Hecke operator \(T_n\) of the given modular form.

HeckePolynomial(M, n : parameters): ModFrm, RngIntElt -> RngUPolElt

Proof: BoolElt                    Default: true

The characteristic polynomial of the \(n\)th Hecke operator \(T_n\). 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 \(q\)th Atkin-Lehner involution \(W_q\) on \(M\) with respect to Basis(M). (Currently \(M\) must be a cuspidal space of modular forms with trivial character and integral weight \(\geq 2\).)

AtkinLehnerOperator(q, f): RngIntElt, ModFrmElt -> ModFrmElt

The image under the involution \(w_q\) of the given modular form.

Example: Hecke Polynomials (ex-f79385)

First we compute a characteristic polynomial on \(S_2(\Gamma_1(13))\) over both \({\mathbb{Z}}\) and the finite field \({\mathbb{F}}_2\).

> 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>
]

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Next we compute a Hecke operator on \(M_4(\Gamma_0(14))\).

> 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)

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This example demonstrates the Atkin-Lehner involution \(W_3\) on \(S_2(\Gamma_0(33))\).

> 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)

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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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