  Let $M$ be a once-punctured torus bundle over $S^1$ with monodromy $h$.
  We show that, under certain hypotheses on $h$, "most" Dehn-fillings of $M$
(in some cases all but finitely many) are virtually
  $\mathbb{Z}$-representable. We apply our results to show that surgeries on
the figure-eight knot with even numerator are virtually
$\mathbb{Z}$-representable.
