For a prime $p\equiv 3$ $(\text{ mod }4)$ and $m\ge 2$, Romik raised a question about whether the Taylor coefficients around $\sqrt{ -1}$ of the classical Jacobi theta function $\theta_3$ eventually vanish modulo $p^m$. This question can be extended to a class of modular forms of half-integral weight on $\Gamma_1(4)$ and CM points; in this paper, we prove an affirmative answer to it for primes $p\ge5$. Our result is also a generalization of the results of Larson and Smith for modular forms of integral weight on $\mathrm{ SL}_2(\mathbb{ Z})$.
Categories: math.NT