Good-prime and bad-prime Hecke coefficient formula ($a_n(T_{p}f)=a_{pn}+\chi (p)p^{k-1}{a}_{n/p}$)

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 $U_p$.