Modular forms of half-integral weight on Γ0(4) with few nonvanishing coefficients modulo ℓ

Let be a nonnegative integer. Let be a number field and be the ring of integers of . Let be a prime and be a prime ideal of over . Let be a modular form of weight on (4) such that its Fourier coefficients are in . In this article, we study sufficient conditions that if has the form with square-free integers , then is congruent to a linear combination of iterated derivatives of a single theta function modulo