On the computation of local components of a newform

We present an algorithm for computing the $p$-component of the automorphic representation arising from a cuspidal newform $f$ for a prime $p$. This is equivalent to computing the restriction to the decomposition group at $p$ of the $\ell$-adic Galois representations attached to $f$ for any $\ell\neq p$. The situation is most interesting when $p^2$ divides the level of $f$, in which case the $p$-component could be supercuspidal. In the supercuspidal case, the local component is induced from an irreducible character of a compact-mod-center subgroup of $\text{GL}_2(\mathbf{Q}_p)$; our algorithm outputs both the group and the irreducible character. We provide examples which illustrate how the local Galois representation can be completely read off from the local component.