  It is well-known that there are automorphic eigenfunctions on
SL(2,Z)\SL(2,R)/SO(2,R) -- such as the classical $j$-function -- that have
exponential growth and have exponentially growing Fourier coefficients (e.g.,
negative powers of $q=e^{2\pi i z}$, or an I-Bessel function). We show that
this phenomenon does not occur on the quotient SL(3,Z)\SL(3,R)/SO(3,R) and
eigenvalues in general position (a removable technical assumption).
  More precisely, if such an automorphic eigenfunction has at most exponential
growth, it cannot have non-decaying Whittaker functions in its Fourier
expansion. This confirms part of a conjecture of Miatello and Wallach, who
assert all automorphic eigenfunctions on this quotient (among other rank $\ge$
2 examples) always have moderate growth. We additionally confirm their
conjecture under certain natural hypotheses, such as the absolute convergence
of the eigenfunction's Fourier expansion.
