Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 137 / 2 / b / Solution

Extend the slash operator for modular forms to positive-determinant matrices byThe double cosethas left-coset representativesTherefore the sum of the corresponding slashes, multiplied by $p^{k/2-1}$, is exactly

Right multiplication by an element of $\Gamma (1)$ permutes these left cosets. The cocycle law for the slash operator consequently givesso $T_p(f)$ is a weight-$k$ level-one modular function. Each displayed summand is holomorphic on $\mathfrak{h}$, hence so is their finite sum. This is the Hecke operator on modular forms.