  We study Fourier term modules on $\mathrm{SU}(2,1)$, which are the modules
arising in Fourier expansions of automorphic forms. Maximal unipotent subgroups
$N$ of $\mathrm{SU}(2,1)$ are non-abelian, and we consider the ``abelian''
Fourier term modules connected to characters of $N$, and also the
``non-abelian'' modules described with theta functions. Poincar\'e series for
$\mathrm{SU}(2,1)$ have in general exponential growth. To deal with such
generalized automorphic forms we allow exponential growth for the functions in
Fourier term modules. We give a complete description of the submodule structure
of all Fourier term modules, and discuss the consequences for Fourier
expansions of automorphic forms.
