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

Let k be a nonnegative integer. Let K be a number field and OK be the ring of integers of K. Let ℓ ≥ 5 be a prime and v be a prime ideal of OK over ℓ. Let f be a modular form of weight k + 1/2 on Γ 0(4) such that its Fourier coefficients are in OK. In this article, we study sufficient conditions that if f has the form f(z)Ξ∑∞ n=1 ∑ t i=1 af(sin2)q sin2 (mod v) with square-free integers si, then f is congruent to a linear combination of iterated derivatives of a single theta function modulo v.