Counting newforms with prescribed ramified supercuspidal components

We give a formula for the number of newforms in $S_k^{\mathrm{n}\mathrm{e}\mathrm{w}}(N)$ 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 $S_k^{\mathrm{n}\mathrm{e}\mathrm{w}}(N)$. It depends only upon the weight, the level, the ramified quadratic extensions $E_p\mathrm{/}{\mathbb{Q}}_p$ 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 $N$.