  Let $\rho$ be a two-dimensional even Galois representation which is induced
from a character $\chi$ of odd order of the absolute Galois group of a real
quadratic field. After imposing some additional conditions on $\chi$, we attach
$\rho$ to a Hecke eigenclass in the cohomology of ${\rm GL}(2,\mathbb Z)$ with
coefficients in a certain infinite-dimensional vector space over a field of
characteristic not equal to 2.
