  Let $\theta$ be an elementary theta function, such as the classical Jacobi
theta function. We establish a spectral decomposition and surprisingly strong
asymptotic formulas for $\langle |\theta|^2, \varphi \rangle$ as $\varphi$
traverses a sequence of Hecke-translates of a nice enough fixed function. The
subtlety is that typically $|\theta|^2 \notin L^2$. Applications to the
subconvexity, quantum variance and $4$-norm problems are indicated.
