  We obtain new lower bounds for the number of Fourier coefficients of a weakly
holomorphic modular form of half-integral weight not divisible by some prime
$\ell$. Among the applications of this we show that there are $\gg
\sqrt{X}/\log \log X$ integers $n \leq X$ for which the partition function
$p(n)$ is not divisible by $\ell$, and that there are $\gg \sqrt{X}/\log \log
X$ values of $n \leq X$ for which $c(n)$, the $n$th Fourier coefficient of the
$j$-invariant, is odd.
