  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.
