  We obtain Gabor frame characterisations of modulation spaces defined via a
class of translation-modulation invariant Banach spaces of distributions that
was recently introduced in $[10]$. We show that these spaces admit an atomic
decomposition through Gabor expansions and that they are characterised by
summability properties of their Gabor coefficients. Furthermore, we construct a
large space of admissible windows. This generalises several fundamental results
for the classical modulation spaces $ M^{p,q}_{w}$. Due to the absence of
solidity assumptions on the Banach spaces defining these modulation spaces, the
methods used for the spaces $M^{p,q}_{w}$ (or, more generally, in coorbit space
theory) fail in our setting and we develop here a new approach based on the
twisted convolution.
