Bad-prime eigenvalue of a trivial-character newform ($a_p=0(p^2\mid M),a_p^2=p^{k-2}(p\parallel M)$)

For a newform of weight $k\ge 2$, level $M$ and trivial Dirichlet character, the bad-prime Hecke operator $U_p$ has the displayed possibilities. The local trace relation on the new subspace of cusp forms gives $U_p=-p^{k/2-1}{W}_p$ when $p\parallel M$, where the normalized local Atkin-Lehner operator squares to one; if $p^2\mid M$, the new-vector relation gives $U_p=0$. In particular $|a_p|<p^{k/2}$, a strict inequality useful for proving irreducibility of old blocks.