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The theta series $\vartheta \left(z\right)=\sum exp\left(2\pi \mathrm{i}{n}^{2}z\right)$ is a classical example of a modular form. In this article we argue that the trace ${\vartheta }_P\left(z\right)=Trexp\left(2\pi \mathrm{i}{P}^{2}z\right)$, where $P$ is a self-adjoint elliptic pseudo-differential operator of order 1 with periodic bicharacteristic flow, may be viewed as a natural generalization. In particular, we establish approximate functional relations under the action of the modular group. This allows a detailed analysis of the asymptotics of ${\vartheta }_P\left(z\right)$ near the real axis, and the proof of logarithm laws and limit theorems for its value...