Periodicities for Taylor coefficients of half-integral weight modular forms

Congruences of Fourier coefficients of modular forms have long been an object of central study.
By comparison, the arithmetic of other expansions of modular forms, in particular Taylor expansions
around points in the upper-half plane, has been much less studied. Recently, Romik made a conjecture
about the periodicity of coefficients around $\tau_0=i$ of the classical Jacobi theta function $\theta_3$.
Here, we  generalize the phenomenon observed by Romik to a broader class of modular forms of
half-integral weight and, in particular, prove the conjecture.

This is joint work with Pavel Guerzhoy and Larry Rolen.