  Let $f\in S_{k+1/2}(N,\chi)$ be a Hecke eigenform of half integral weight
$k+1/2\,(k\geq 2)$ and the real nebentypus $\chi=\pm 1$ where the Fourier
coefficients $a(n)$ are reals. We prove that the sequence
$\{\chi(p^{\nu})a(tp^{2\nu})\}_{\nu\in\N}$ has infinitely many sign changes for
almost all primes $p$ where $t$ is a squarefree integer such that $a(t)\neq 0$.
The same result holds for the sequences of Fourier coefficients
$\{a(tp^{2(2\nu+1)})\}_{\nu\in\N}$ and $\{a(tp^{4\nu})\}_{\nu\in\N}$.
