Ramanujan's congruence primes

Abstract

Ramanujan showed that $\tau(p) \equiv p^{11}+1 \pmod{691}$, where $\tau(n)$ is the $n$-th Fourier coefficient of the unique normalized cusp form of weight $12$ and full level, and the prime $691$ appears in the numerator of $\zeta(12)/\pi^{12}$ for the Riemann zeta function $\zeta(s)$. Searching for such congruences, it is shown that the prime $67$ appears in the numerator of $L(6,\chi)/(\pi^6 \sqrt{5})$, where $\chi$ is the unique nontrivial quadratic Dirichlet character modulo $5$ and $L(s,\chi)$ its Dirichlet $L$-function, giving rise to a congruence $f_\chi \equiv E^\circ_{6, \chi} \pmod{67}$ between a cusp form $f_\chi$ and an Eisenstein series $E^\circ_{6, \chi}$ of weight $6$ on $\Gamma_0(5)$ with nebentypus character $\chi.$