  Let $M$ be a positive integer and $p(n)$ be the number of partitions of a
positive integer $n$. Newman's conjecture asserts that for each integer $r$,
there are infinitely many positive integers $n$ such that \[ p(n)\equiv r
\pmod{M}. \] For an integer $d$, let $B_{d}$ be the set of positive integers
$M$ such that the number of prime divisors of $M$ is $d$. In this paper, we
prove that for each positive integer $d$, the density of the set of positive
integers $M$ for which Newman's conjecture holds in $B_{d}$ is $1$.
Furthermore, we study an analogue of Newman's conjecture for weakly holomorphic
modular forms on $\Gamma_0(N)$ with nebentypus, and this applies to $t$-core
partitions and generalized Frobenius partitions with $h$-colors.
