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 $\mathrm{\ell }$-adic Galois representations attached to $f$ for any $\mathrm{\ell }\mathrm{\ne }p$. 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.