Counting newforms with prescribed ramified supercuspidal components

Abstract

We give a formula for the number of newforms in $S_k^{\mathrm{new}}(N)$ that have prescribed ramified supercuspidal components $\pi_p$ at a set $T$ of primes dividing $N$. This dimension is given in terms of the trace of the Atkin--Lehner operator at $T$ on $S_k^{\mathrm{new}}(N)$. It depends only upon the weight, the level, the ramified quadratic extensions $E_p/{\mathbb Q}_p$ attached to the $\pi_p$, and the root number of each $\pi_p$. The formula is completely explicit when $T$ consists of either a single prime or all prime factors of $N$.