A generalization of Ramanujan's congruence to modular forms of prime level

Abstract

We prove congruences between cuspidal newforms and Eisenstein series of prime level, which generalize Ramanujan's congruence. Such congruences were recently found by Billerey and Menares, and we refine them by specifying the Atkin-Lehner eigenvalue of the newform involved. We show that similar refinements hold for the level raising congruences between cuspidal newforms of different levels, due to Ribet and Diamond. The proof relies on studying the new subspace and the Eisenstein subspace of the space of period polynomials for the congruence subgroup $\Gamma_0(N)$, and on a version of Ihara's lemma.