The spectral decomposition of $\mathrm{\mid }\theta {\mathrm{\mid }}^2$

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 $⟨\mathrm{\mid }\theta {\mathrm{\mid }}^2,\phi ⟩$ as $\phi$ traverses a sequence of Hecke-translates of a nice enough fixed function. The subtlety is that typically $\mathrm{\mid }\theta {\mathrm{\mid }}^2\mathrm{\notin }{L}^2$. Applications to the subconvexity, quantum variance and $4$-norm problems are indicated.