Locally potentially equivalent Galois representations

We show that if two continuous semi-simple \(\ell \)-adic Galois representations are locally potentially equivalent at a sufficiently large set of places then they are globaly potentially equivalent. We also prove an analogous result for arbitrarily varying powers of character values evaluated at the Frobenius conjugacy classes. In the context of modular forms, we prove: given two non-CM newforms $f$ and $g$ of weight at least two, such that $a_p(f)^{n_p}=a_p(g)^{n_p}$ on a set of primes of positive upper density and for some set of natural numbers $n_p$, then $f$ and $g$ are twists of each other by a Dirichlet character.