$p$-adic Properties for Taylor Coefficients of Half-integral Weight Modular Forms on $\Gamma_1(4)$

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