Fourier coefficients of a composite-index Hecke operator
= Fourier coefficients of a composite-index Hecke operator
{c}
{title2=$a_r(T_nf)=\sum_{a\mid\gcd(n,r)}a^{k-1}a_{nr/a^2}(f)$}
For the determinant-normalized <slash operator for modular forms>, $T_nf=n^{k/2-1}\sum_{\gamma\in\Pi_n}f|_k\gamma$. If $f=\sum_{r\geq0}a_rq^r$, then $a_r(T_nf)=\sum_{a\mid\gcd(n,r)}a^{k-1}a_{nr/a^2}(f)$, with $\gcd(n,0)=n$. In particular $a_1(T_nf)=a_n(f)$, integer coefficients are preserved, and <cusp forms> remain cusp forms.