Let F be a field of characteristic not equal to 2. Let r be a 2-dimensional even Galois representation which is induced from an F-valued character of odd order of the absolute Galois group of a real quadratic field K. After imposing some additional conditions on the inducing character c, we attach r to a Hecke eigenclass in the cohomology of GL(2,Z) with coefficients in a certain countably-infinite-dimensional vector space over F. We hope that this work will enable us to prove some cases of a Serre-type conjecture for certain reducible 3-dimensional Galois representations. This is joint work with Darrin Doud.