  Recently, Bruinier and Ono classified cusp forms $f(z) := \sum_{n=0}^{\infty}
a_f(n)q ^n \in S_{\lambda+1/2}(\Gamma_0(N),\chi)\cap \mathbb{Z}[[q]]$ that does
not satisfy a certain distribution property for modulo odd primes $p$. In this
paper, using Rankin-Cohen Bracket, we extend this result to modular forms of
half integral weight for primes $p \geq 5$. As applications of our main theorem
we derive distribution properties, for modulo primes $p\geq5$, of traces of
singular moduli and Hurwitz class number. We also study an analogue of Newman's
conjecture for overpartitions.
