Congruences and ramified primes in fields of coefficients of newforms

Abstract

We investigate the splitting behavior of $\ell$ in the coefficient field of a newform $f$ of level $N$, under the assumption that $f$ is congruent modulo a prime above $\ell$ to another newform $g$ whose level divides $N/p^2$ for some prime $p\mid N$. In particular, we show that the maximal real subfield of the $\ell$-th cyclotomic field, $\mathbb{Q}(\zeta_\ell + \zeta_\ell^{-1})$, is contained in the coefficient field of $f$. We conclude by presenting explicit examples that illustrate these results.