Congruences of the partition function

Let $p(n)$ denote the partition function. In this article, we will show that congruences of the form $$p(m^j{\mathrm{\ell }}^{k}n+B)\equiv 0\mathrm{m}\mathrm{o}\mathrm{d}m\text{for all}n\ge 0$$ exist for all primes $m$ and $\mathrm{\ell }$ satisfying $m\ge 13$ and $\ell\neq 2,3,m$\mathrm{.}Heretheinteger$k$ depends on the Hecke eigenvalues of a certain invariant subspace of $S_{m\mathrm{/}2-1}({\mathrm{\Gamma }}_0(576),{\chi }_{12})$ 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 $$p(m^j{\mathrm{\ell }}^{k}n+B)\equiv 0\mathrm{m}\mathrm{o}\mathrm{d}{m}^i$$ holds for all integers $n$ not divisible by $\mathrm{\ell }$.