Hecke operator

Remarks

If $f = \sum_{m \ge 0} c(m) q^m$ with $q = e^{2 \pi i \tau}$, then $T_n f = \sum_{m \ge 0} \gamma(m) q^m$ with $$\gamma(m) = \sum_{d \mid \gcd(m, n)} d^{k-1} c(mn/d^2).$$ The operators commute, $T_m T_n = T_{mn}$ when $\gcd(m, n) = 1$, and $T_p T_{p^r} = T_{p^{r+1}} + p^{k-1} T_{p^{r-1}}$ for primes $p$ and $r \ge 1$. They map the cusp forms $S_k$ into $S_k$. Example: $S_{12} = \C \Delta$ is one-dimensional, so $T_n \Delta$ is a multiple of $\Delta$; comparing the coefficients of $q$ gives $T_n \Delta = \tau(n) \Delta$. Non-example: the map $f(\tau) \mapsto f(n\tau)$ alone does not preserve $M_k$. For instance, $E_4(2\tau) = 1 + 240 q^2 + \cdots$ is not a multiple of $E_4 = 1 + 240 q + \cdots$, while $M_4 = \C E_4$.