An evident corollary arising from Newton-Thorne

Abstract

Let $f$ be a primitive holomorphic newform of weight at least $2$ and arbitrary level. We consider the representations obtained from its symmetric powers by taking tensor products, symmetric powers, and isobaric sums. Classical $\mathrm{GL}(2)$ representation theory decomposes each such expression into symmetric powers with determinant twists. This gives an automorphic realization and a factorization of the associated $L$-function. At a prime dividing the level, the same decomposition must be applied to the full Weil-Deligne parameter before inertia invariants and the kernel of monodromy are taken. For elliptic curves over $\mathbb{Q}$, we record the resulting local factors, conductors, and root numbers using the calculations of Dummigan-Martin-Watkins and Martin-Watkins.