The root lattice $A^*_n$ and Ramanujan's circular summation of theta functions

We relate a formula of Ramanujan on the circular summation of the $n$th power of theta functions, $F_n(q)$, to the theta series of the root lattice $A^*_n$. We then use properties of the lattice to show that $F_n$ includes an $\operatorname{SL}_2(\mathbf{Z})$ modular form when $n$ is an odd perfect square as well as to derive a very simple expression for $F_9(q)$.