  Suppose that $G$ is a simple reductive group over $\mathbf{Q}$, with an
exceptional Dynkin type, and with $G(\mathbf{R})$ quaternionic (in the sense of
Gross-Wallach). In a previous paper, we gave an explicit form of the Fourier
expansion of modular forms on $G$ along the unipotent radical of the Heisenberg
parabolic. In this paper, we give the Fourier expansion of the minimal modular
form $\theta_{Gan}$ on quaternionic $E_8$, and some applications. The
$Sym^{8}(V_2)$-valued automorphic function $\theta_{Gan}$ is a weight four,
level one modular form on $E_8$, which has been studied by Gan. The
applications we give are the construction of special modular forms on
quaternionic $E_7, E_6$ and $G_2$. We also discuss a family of degenerate
Heisenberg Eisenstein series on the groups $G$, which may be thought of as an
analogue to the quaternionic exceptional groups of the holomorphic Siegel
Eisenstein series on the groups $\mathrm{GSp}_{2n}$.
