Suppose that ℓ ≥ 5 \ell \geq 5 is prime. For a positive integer N N with 4 ∣ N 4 \mid N , previous works studied properties of half-integral weight modular forms on Γ 0 ( N ) \Gamma _0(N) which are supported on finitely many square classes modulo ℓ \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 Θ \Theta -operator. Here, we study the analogous problem for modular forms of half-integral weight on SL 2 ⁡ ( Z ) \operatorname {SL}_{2}(\mathbb {Z}) . Let η \eta be the Dedekind eta function. For a wide range of weights, we prove that every half-integral weight modular form on SL 2 ⁡ ( Z ) \operatorname {SL}_{2}(\mathbb {Z}) which is supported on finitely many square classes modulo ℓ \ell can be written modulo ℓ \ell in terms of η ℓ \eta ^{\ell } and an iterated derivative of η \eta .