  We consider converses to the density theorem for irreducible, projective,
unitary group representations restricted to lattices using the dimension theory
of Hilbert modules over twisted group von Neumann algebras. We show that under
the right assumptions, the restriction of a $\sigma$-projective unitary
representation $\pi$ of a group $G$ to a lattice $\Gamma$ extends to a Hilbert
module over the twisted group von Neumann algebra $\text{L}(\Gamma,\sigma)$. We
then compute the center-valued von Neumann dimension of this Hilbert module.
For abelian groups with 2-cocycle satisfying Kleppner's condition, we show that
the center-valued von Neumann dimension reduces to the scalar value $d_{\pi}
\text{vol}(G/\Gamma)$, where $d_{\pi}$ is the formal dimension of $\pi$ and
$\text{vol}(G/\Gamma)$ is the covolume of $\Gamma$ in $G$. We apply our results
to characterize the existence of multiwindow super frames and Riesz sequences
associated to $\pi$ and $\Gamma$. In particular, we characterize when a lattice
in the time-frequency plane of a second countable, locally compact abelian
group admits a Gabor frame or Gabor Riesz sequence.
