  Quaternionic modular forms on the split exceptional group $G_2 = G_2^s$ were
defined by Gan-Gross-Savin. A remarkable property of these automorphic
functions is that they have a robust notion of Fourier expansion and Fourier
coefficients, similar to the classical holomorphic modular forms on Shimura
varieties. In this paper we prove that in even weight $\ell$ at least $6$,
there is a basis of the space of cuspidal modular forms of weight $\ell$ such
that all the Fourier coefficients of elements of this basis are in the
cyclotomic extension of $\mathbf{Q}$. Our main tool for proving this is to
develop a notion of "exceptional theta functions" on $G_2$.
