  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 $\eta$.
