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 finitely many square classes modulo $\ell$, in some cases proving these congruent to the image single variable theta series under number iterations Ramanujan $\Theta$-operator. Here, we study analogous problem for $\operatorname{SL}_{2}(\mathbb{Z})$. Let $\eta$ be Dedekind eta function. wide range weights, prove every form $\operatorname{SL}_{2}(\mathbb{Z})$ $\ell$ can written terms $\eta^{\ell}$ and an iterated derivative $\eta$.