Fourier coefficients of a composite-index Hecke operator ($a_r(T_{n}f)={\sum }_{a\mid \gcd (n,r)}{a}^{k-1}{a}_{nr/a^2}(f)$)

For the determinant-normalized slash operator for modular forms, $T_{n}f=n^{k/2-1}{\sum }_{\gamma \in {\Pi }_n}f{|}_k\gamma$. If $f={\sum }_{r\ge 0}{a}_r{q}^r$, then $a_r(T_{n}f)={\sum }_{a\mid \gcd (n,r)}{a}^{k-1}{a}_{nr/a^2}(f)$, with $\gcd (n,0)=n$. In particular $a_1(T_{n}f)=a_n(f)$, integer coefficients are preserved, and cusp forms remain cusp forms.