Ramanujan-style congruences for prime level

We establish Ramanujan-style congruences modulo certain primes ℓ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell $$\end{document} between an Eisenstein series of weight k, prime level p and a cuspidal newform in the ε\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon $$\end{document}-eigenspace of the Atkin–Lehner operator inside the space of cusp forms of weight k for Γ0(p)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma _0(p)$$\end{document}. 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.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell .$$\end{document} The degree of the number field generated by these coefficients we investigate using recent results on prime factors of shifted prime numbers.