We consider the problem of constructing ?projectively rational? lifts of odd, two-dimensional Galois representations with values in ?Using modular forms, in particular, the theory of congruences, we compute such lifts for many examples of mod 7 representations thus giving evidence that suggests that such lifts may always exist. We also consider the invariance after twist (weight change) of the existence of such lifts.

projectively rational, Galois representations, modularity conjecture.