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

TL;DR

This paper extends Ramanujan's congruence to modular forms of prime level by establishing new congruences between cuspidal newforms and Eisenstein series, refining previous results with Atkin-Lehner eigenvalues.

Contribution

It generalizes Ramanujan's congruence to prime level modular forms and refines existing congruences by incorporating Atkin-Lehner eigenvalues and level raising scenarios.

Findings

Proves new congruences between cuspidal newforms and Eisenstein series of prime level.

Refines previous congruences by specifying Atkin-Lehner eigenvalues.

Extends congruences to level raising between forms of different levels.

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.