Congruences for Level $1$ cusp forms of half-integral weight

Suppose that $\mathrm{\ell }\ge 5$ is prime. For a positive integer $N$ with $4 \mid N$,previousworksstudiedpropertiesofhalf-integralweightmodularformson$\Gamma_0(N)$whicharesupportedonfinitelymanysquareclassesmodulo$\ell$, in some cases proving that these forms are congruent to the image of a single variable theta series under some number of iterations of the Ramanujan $\mathrm{\Theta }$-operator. Here, we study the analogous problem for modular forms of half-integral weight on ${\mathrm{SL}}_2(\mathbb{Z})$. Let $\eta$ be the Dedekind eta function. For a wide range of weights, we prove that every half-integral weight modular form on ${\mathrm{SL}}_2(\mathbb{Z})$ which is supported on finitely many square classes modulo $\mathrm{\ell }$ can be written modulo $\mathrm{\ell }$ in terms of ${\eta }^{\mathrm{\ell }}$ and an iterated derivative of $\eta$.