Theta Cycles of Modular Forms Modulo $p^2$

The theta cycle of a modular form modulo a prime $p\geq 5$ is well understood. By contrast, the theta cycle modulo a power of $p$ is still mysterious and experimentally erratic. Here we completely determine the theta cycle of a weight $k < p$ modular form modulo $p^2$ on the initial segment of length $p$ and we prove exact values or nontrivial bounds for the weight filtrations on $p-2$ further segments of length $p - k + 1$. In particular, asymptotically as $p \to \infty$ we establish 50% of the theta cycle exactly, and we provide nontrivial bounds for 100% of it. We determine the first two low points exactly and $\left\lfloor \frac{p - k + 1}{2} \right\rfloor$ further low points at regular positions. Moreover, we detect low points at exceptional positions which solve a quadratic equation modulo $p$, and which disturb the otherwise regular structure in the segments that we exhibit.