Ramanujan's partition generating functions modulo $\mathrm{\ell }$

For the partition function $p(n)$, Ramanujan proved the striking identities $$P_5(q):=\sum _{n\ge 0}p(5n+4)q^n=5\prod _{n\ge 1}\frac{{(q^5;q^5)}_{\mathrm{\infty }}^5}{(q;q{)}_{\mathrm{\infty }}^6},$$ $$P_7(q):=\sum _{n\ge 0}p(7n+5)q^n=7\prod _{n\ge 1}\frac{{(q^7;q^7)}_{\mathrm{\infty }}^3}{(q;q{)}_{\mathrm{\infty }}^4}+49q\prod _{n\ge 1}\frac{{(q^7;q^7)}_{\mathrm{\infty }}^7}{(q;q{)}_{\mathrm{\infty }}^8},$$ where $(q;q{)}_{\mathrm{\infty }}:={\prod }_{n\ge 1}(1-q^n)\mathrm{.}$ As these identities imply his celebrated congruences modulo 5 and 7, it is natural to seek, for primes $\mathrm{\ell }\ge 5,$ closed form expressions of the power series $$P_{\mathrm{\ell }}(q):=\sum _{n\ge 0}p(\mathrm{\ell }n-{\delta }_{\mathrm{\ell }})q^n(\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{\ell }),$$ where ${\delta }_{\mathrm{\ell }}:=\frac{{\mathrm{\ell }}^2-1}{24}\mathrm{.}$ In this paper, we prove that $$P_{\mathrm{\ell }}(q)\equiv c_{\mathrm{\ell }}\frac{T_{\mathrm{\ell }}(q)}{(q^{\mathrm{\ell }};q^{\mathrm{\ell }}{)}_{\mathrm{\infty }}}(\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{\ell }),$$ where $c_{\mathrm{\ell }}\in \mathbb{Z}$ is explicit and $T_{\mathrm{\ell }}(q)$ is the generating function for the Hecke traces of $\mathrm{\ell }$-ramified values of special Dirichlet series for weight $\mathrm{\ell }-1$ cusp forms on $S{L}_2(\mathbb{Z})$. This is a new proof of Ramanujan’s congruences modulo 5, 7, and 11, as there are no nontrivial cusp forms of weight 4, 6, and 10.