The spectral decomposition of |θ|2

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