  We derive major parts of the eigenvalue spectrum of the operators on the
squashed seven-sphere that appear in the compactification of eleven-dimensional
supergravity. These spectra determine the mass spectrum of the fields in
$AdS_4$ and are important for the corresponding ${\mathcal N} =1$
supermultiplet structure. This work is a continuation of the work in [1] where
the complete spectrum of irreducible isometry representations of the fields in
$AdS_4$ was derived for this compactification. Some comments are also made
concerning the $G_2$ holonomy and its implications on the structure of the
operator equations on the squashed seven-sphere.
