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

### Robert Dicks – University Illinois Urbana-Champaign

Suppose that *ℓ*≥5 is prime. For a positive integer *N* with 4∣*N*, previous works studied properties of half-integral weight modular forms on Γ0(*N*) which are supported on finitely many square classes modulo *ℓ*, 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-operator. Here, we study the analogous problem for modular forms of half-integral weight on SL2(ℤ). Let *η* be the Dedekind eta function. For a wide range of weights, we prove that every half-integral weight modular form on SL2(ℤ) which is supported on finitely many square classes modulo *ℓ* can be written modulo *ℓ* in terms of *ηℓ* and an iterated derivative of *η*.