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

Suppose that $\ell \geq 5$ is prime. For a positive integer $N$ with $4 \mid N$, previous works studied properties of half-integral weight modular forms on $\Gamma_0(N)$ which are supported on finitely many square classes modulo $\ell$, 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 $\Theta$-operator. Here, we study the analogous problem for modular forms of half-integral weight on $\operatorname{SL}_{2}(\mathbb{Z})$. Let $\eta$ be the Dedekind eta function. For a wide range of weights, we prove that every half-integral weight modular form on $\operatorname{SL}_{2}(\mathbb{Z})$ which is supported on finitely many square classes modulo $\ell$ can be written modulo $\ell$ in terms of $\eta^{\ell}$ and an iterated derivative of $\eta$.