I will try to explain what the Fourier expansion of a "modular form" on an exceptional group looks like, from the point of view of the archimedean place.  In more detail, Gross-Wallach and Gan-Gross-Savin have singled out what a modular form on an exceptional group G should be: The real points G(R) should make up the so-called quaternionic real form of G, and then modular forms F on G correspond to automorphic forms whose infinite component belongs to the quaternionic discrete series.  In such a situation, the Fourier expansion of F is controlled by what is called generalized Whittaker function.  Wallach has studied these functions, and proved (abstractly) that they satisfy a finite multiplicity statement. When this finite multiplicity is 1, it makes sense to ask for a formula for the generalized Whittaker function.  I will give a formula in the above setting.