Distinguishing Galois representations by their normalized traces

Suppose $ρ_1$ and $ρ_2$ are two pure Galois representations of the absolute Galois group of a number field $K$ of weights $k_1$ and $k_2$ respectively, having equal normalized Frobenius traces $Tr(ρ_1(σ_v)) /Nv^{k_1/2}$ and $Tr(ρ_2(σ_v)) /Nv^{k_2/2}$ at a set of primes $v$ of $K$ with positive upper density. Assume further that the algebraic monodromy group of $ρ_1$ is connected and the repesentation is absolutely irreducible. We prove that $ρ_1$ and $ρ_2$ are twists of each other by power of a Tate twist times a character of finite order. We apply this to modular forms and deduce a result proved by Murty and Pujahari.

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