  In this paper we analyze Fourier coefficients of automorphic forms on a
finite cover $G$ of an adelic split simply-laced group. Let $\pi$ be a minimal
or next-to-minimal automorphic representation of $G$. We prove that any
$\eta\in \pi$ is completely determined by its Whittaker coefficients with
respect to (possibly degenerate) characters of the unipotent radical of a fixed
Borel subgroup, analogously to the Piatetski-Shapiro--Shalika formula for cusp
forms on $GL_n$. We also derive explicit formulas expressing the form, as well
as all its maximal parabolic Fourier coefficient in terms of these Whittaker
coefficients. A consequence of our results is the non-existence of cusp forms
in the minimal and next-to-minimal automorphic spectrum. We provide detailed
examples for $G$ of type $D_5$ and $E_8$ with a view towards applications to
scattering amplitudes in string theory.
