  We calculate the low-energy spectral weight of a holographic superfluid
coupled to a Chern-Simons term in IR radial scaling geometries characterized by
a parameter $\eta$. This work was motivated by previous results where an
unexpected low-energy spectral weight and a region of instability were seen,
both at finite momentum, for the holographic superfluid. We characterize the
effect of varying the Chern-Simons coupling $\alpha$ and condensate charge
parameter $\zeta$ on these regions supporting low-energy spectral weight or a
finite momentum instability. We show that $\eta$, $\alpha$ and $\zeta$ each
plays a unique role in shaping these regions. We find a surface
$\alpha_{\text{crit}}(\eta, \zeta)$ above which the theory is unstable. In the
longitudinal channel we extend our analysis to general dimension $d$. We
briefly analyze the Einstein-Maxwell-dilaton theory and find that Fermi shells
exist for $d>4$.
