  We study the question of Eulerianity (factorizability) for Fourier
coefficients of automorphic forms, and we prove a general transfer theorem that
allows one to deduce the Eulerianity of certain coefficients from that of
another coefficient. We also establish a `hidden' invariance property of
Fourier coefficients. We apply these results to minimal and next-to-minimal
automorphic representations, and deduce Eulerianity for a large class of
Fourier and Fourier-Jacobi coefficients. In particular, we prove Eulerianity
for parabolic Fourier coefficients with characters of maximal rank for a class
of Eisenstein series in minimal and next-to-minimal representations of groups
of ADE-type that are of interest in string theory.
