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Operators








Each space M of modular forms comes equipped with a commuting family
T_1, T_2, T_3, ... 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 T_n 
with respect to Basis(M).
(Currently M must be a space  of modular forms 
with trivial character and integral weight >= 2.)



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



    Proof: BoolElt                      Default: true



The characteristic polynomial of the nth 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 arbitrary.



AtkinLehnerOperator(M, q) : ModFrm, RngIntElt -> AlgMatElt



The matrix representing the qth 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 >= 2.)





Example ModFrm_HeckePolynomials (H97E13)

First we compute a characteristic polynomial on S_2(Gamma_1(13))
over both Z and the finite field 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>
]


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)




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)


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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