 Solution

For $L_{\tau }=\mathbb{Z}\tau +\mathbb{Z}$, the index-$p$ overlattices areThe first $p$ preserve the marked point $1/N$. At a good prime the last equals $p^{-1}{L}_{p\tau }$, with marked point corresponding after scaling to $p/N$. Consequently the lattice formula becomesThe first term has coefficient $a_{pn}$, because the sum of $p$th roots of unity is zero unless its exponent is divisible by $p$. The second has coefficient $\chi (p)p^{k-1}{a}_{n/p}$ when $p\mid n$ and zero otherwise. ThusThis includes $b_0=(1+\chi (p)p^{k-1})a_0$ at a good prime.

At $p\mid N$, the last overlattice loses the required exact order and is excluded. The operator is then $U_p$, with $b_n=a_{pn}$. The formula remains valid for all primes if the Dirichlet character is extended to integers by zero on nonunits, so $\chi (p)=0$ at bad primes. Without that extension, the printed $\chi (p)$ is defined only when $p\nmid N$. This is the good-prime and bad-prime Hecke coefficient formula.