Ramanujan-style congruences for prime level

We establish Ramanujan-style congruences modulo certain primes $$\ell $$ between an Eisenstein series of weight k, prime level p and a cuspidal newform in the $$\varepsilon -eigenspace Atkin–Lehner operator inside space cusp forms k for $$\Gamma _0(p)$$ . Under mild assumption, this refines result Gaba–Popa. use these recent work Ciolan, Languasco third author on Euler–Kronecker constants, to quantify non-divisibility Fourier coefficients involved by .$$ The degree number field generated we investigate using results factors shifted numbers.