Distinguishing Galois representations by their normalized traces

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 $Tr({\rho }_1({\sigma }_v))\mathrm{/}N{v}^{k_1\mathrm{/}2}$ and $Tr({\rho }_2({\sigma }_v))\mathrm{/}N{v}^{k_2\mathrm{/}2}$ 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.