Recovering l-adic representations

We consider the problem of recovering l-adic representations from a knowledge of the character values at the Frobenius elements associated to l-adic representations constructed algebraically out of the original representations. These results generalize earlier results in of the author concerning refinements of strong multiplicity one for $l$-adic represntations, and a result of Ramakrishnan recovering modular forms from a knowledge of the squares of the Hecke eigenvalues. For example, we show that if the characters of some tensor or symmetric powers of two absolutely irreducible l-adic representation with the algebraic envelope of the image being connected, agree at the Frobenius elements corresponding to a set of places of positive upper density, then the representations are twists of each other by a finite order character.