Ramanujan-style congruences for prime level

Abstract

We establish Ramanujan-style congruences modulo certain primes ℓ between an Eisenstein series of weight k, prime level p and a cuspidal newform in the ε-eigenspace of the Atkin–Lehner operator inside the space of cusp forms of weight k for Γ (p). Under a mild assumption, this refines a result of Gaba–Popa. We use these congruences and recent work of Ciolan, Languasco and the third author on Euler–Kronecker constants, to quantify the non-divisibility of the Fourier coefficients involved by ℓ. The degree of the number field generated by these coefficients we investigate using recent results on prime factors of shifted prime numbers. © 2022, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.