= Bad-prime eigenvalue of a trivial-character newform
{title2=$a_p=0\ (p^2\mid M),\qquad a_p^2=p^{k-2}\ (p\parallel M)$}
For a <newform> of weight $k\ge2$, 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.
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