Hecke operator ($T_p$)

For a prime $p$, one normalization of the weight-$k$ Hecke operator at level one isIf $f={\sum }_n{a}_n{q}^n$, thenwhere $a_{n/p}=0$ unless $p$ divides $n$.

Let a level-one modular function be holomorphic on the upper half-plane. If the span of $f,T_{p}f,T_p^{2}f,\dots$ is finite-dimensional, then $f$ is holomorphic at infinity. Indeed, a pole of order $N$ would make $T_p^{r}f$ have pole order $p^{r}N$, producing linearly independent functions.