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For a prime $p\equiv 3$ $(\text{mod }4)$ and $m\ge 2$, Romik raised aquestion about whether the Taylor coefficients around $\sqrt{-1}$ of theclassical Jacobi theta function $\theta_3$ eventually vanish modulo $p^m$. Thisquestion 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 toit for primes $p\ge5$. Our result is also a generalization of the results ofLarson and Smith for modular forms of integral weight on$\mathrm{SL}_2(\mathbb{Z})$.

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