= Solution
For $L_\tau=\mathbb Z\tau+\mathbb Z$, the index-$p$ overlattices are
$$
L_b=\mathbb Z\frac{\tau+b}{p}+\mathbb Z\quad(0\le b<p),\qquad L_\infty=\mathbb Z\tau+\frac1p\mathbb Z.
$$
The 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 becomes
$$
\boxed{T_pf(\tau)=\frac1p\sum_{b=0}^{p-1}f\left(\frac{\tau+b}{p}\right)+p^{k-1}\chi(p)f(p\tau)\quad(p\nmid N).}
$$
The 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. Thus
$$
\boxed{b_n=\begin{cases}a_{pn},&p\nmid n,\\a_{pn}+\chi(p)p^{k-1}a_{n/p},&p\mid n.\end{cases}}
$$
This 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>.
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