Hecke operator on marked lattices ($T_{p}F(L,t)=p^{-1}{\sum }_{L^{\prime }}F(L^{\prime },t\mathrm{mod}{L}^{\prime })$)

Sum over index-$p$ overlattices in which the marked point retains exact order $N$. There are $p+1$ summands at good primes and $p$ at bad primes. With homogeneity $F(uL,ut)=u^{-k}F(L,t)$, the coefficient $1/p$ is the normalization giving the standard Hecke operator Fourier action. Cusp holomorphy under rational slash operators proves preservation of modular forms.

At a prime not dividing the level, the root-of-unity average contributes $a_{pn}$ and the scaling overlattice contributes the second term. At a prime dividing the level that overlattice loses the marked point's exact order and is excluded. Extending the Dirichlet character by zero gives a uniform formula, with $a_{n/p}=0$ unless $p\mid n$. The bad-prime operator is $U_p$.