  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
$\mathrm{SL}_2(\mathbb{Z})$.
