I will report on joint work with Spencer Leslie where we define an analogue of the Cohen-Zagier Eisenstein series to the exceptional group $G_2$. Recall that the Cohen-Zagier Eisenstein series is a weight $3/2$ modular form whose Fourier coefficients see the class numbers of imaginary quadratic fields. We define a particular modular form of weight $1/2$ on $G_2$, and prove that its Fourier coefficients see (certain torsors for) the 2-torsion in the narrow class groups of totally real cubic fields. In particular:

1) we define a notion of modular forms of half-integral weight on certain exceptional groups,
2) we prove that these modular forms have a nice theory of Fourier coefficients, and
3) we partially compute the Fourier coefficients of a particular nice example on $G_2$.