  The purpose of this paper is to collect and make explicit the results of
Gel'fand, Graev and Piatetski-Shapiro and Miyazaki for the $GL(3)$ cusp forms
which are non-trivial on $SO(3,\mathbb{R})$. We give new descriptions of the
spaces of cusp forms of minimal $K$-type and from the Fourier-Whittaker
expansions of such forms give a complete and completely explicit spectral
expansion for $L^2(SL(3,\mathbb{Z})\backslash PSL(3,\mathbb{R}))$, accounting
for multiplicities, in the style of Duke, Friedlander and Iwaniec's paper on
Artin $L$-functions. We directly compute the Jacquet integral for the Whittaker
functions at the minimal $K$-type, improving Miyazaki's computation. The
primary tool will be the study of the differential operators coming from the
Lie algebra on vector-valued cusp forms.
