  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.
