Suppose ( rho_1 ) and ( rho_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(rho_1(sigma_v)) /Nv^{k_1/2}) and ( Tr(rho_2(sigma_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 $rho_1$ is connected and the repesentation is absolutely irreducible. We prove that ( rho_1 ) and ( rho_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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