  Let $p(n)$ denote the partition function. In this article, we will show that
congruences of the form $$ p(m^j\ell^kn+B)\equiv 0\mod m \text{for all} n\ge 0
$$ exist for all primes $m$ and $\ell$ satisfying $m\ge 13$ and $\ell\neq
2,3,m$. Here the integer $k$ depends on the Hecke eigenvalues of a certain
invariant subspace of $S_{m/2-1}(\Gamma_0(576),\chi_{12})$ and can be
explicitly computed. More generally, we will show that for each integer $i>0$
there exists an integer $k$ such that for every non-negative integers $j\ge i$
with a properly chosen $B$ the congruence $$ p(m^j\ell^kn+B)\equiv 0\mod m^i $$
holds for all integers $n$ not divisible by $\ell$.
