mfhecke

Function: mfhecke
Section: modular_forms
C-Name: mfhecke
Prototype: GGL
Help: mfhecke(mf,F,n): F being a modular form in space mf, returns T(n)F,
 where T(n) is the n-th Hecke operator. Warning: if F is of level M<N,
 T(n)F is in general not the same in M_k(G_0(M),CHI) and in M_k(G_0(N),CHI).
 We take T(n) at the same level as the one used in mf.
Doc: $F$ being a modular form in modular form space \var{mf}, returns
 $T(n)F$, where $T(n)$ is the $n$-th Hecke operator.

 \misctitle{Warning} If $F$ is of level $M<N$, then $T(n)F$
 is in general not the same in $M_k(\Gamma_0(M),\chi)$ and in
 $M_k(\Gamma_0(N),\chi)$. We take $T(n)$ at the same level as the one used in
 \kbd{mf}.
 \bprog
 ? mf = mfinit([26,2],0); F = mfbasis(mf)[1]; mftobasis(mf,F)
 %1 = [1, 0]~
 ? G2 = mfhecke(mf,F,2); mftobasis(mf,G2)
 %2 = [0, 1]~
 ? G5 = mfhecke(mf,F,5); mftobasis(mf,G5)
 %3 = [-2, 1]~
 @eprog\noindent Modular forms of half-integral weight are supported, in
 which case $n$ must be a perfect square, else $T_n$ will act as $0$ (the
 operator $T_p$ for $p \mid N$ is not supported yet):
 \bprog
 ? F = mfpow(mfTheta(),3); mf = mfinit(F);
 ? mfisequal(mfhecke(mf,F,9), mflinear([F],[4]))
 %2 = 1
 @eprog ($F$ is an eigenvector of all $T_{p^2}$, with eigenvalue $p+1$ for
 odd $p$.)

 \misctitle{Warning} When $n$ is a large composite, resp.~the square of a large
 composite in half-integral weight, it is in general more efficient to use
 \kbd{mfheckemat} on the \kbd{mftobasis} coefficients:
 \bprog
 ? mfcoefs(mfhecke(mf,F,3^10), 10)
 time = 917 ms.
 %3 = [324, 1944, 3888, 2592, 1944, 7776, 7776, 0, 3888, 9720, 7776]
 ? M = mfheckemat(mf,3^10) \\ instantaneous
 %4 =
 [324]
 ? G = mflinear(mf, M*mftobasis(mf,F));
 ? mfcoefs(G, 10) \\ instantaneous
 %6 = [324, 1944, 3888, 2592, 1944, 7776, 7776, 0, 3888, 9720, 7776]
 @eprog