Congruences and ramified primes in fields of coefficients of newforms

TL;DR

This paper studies how primes split in the coefficient fields of newforms, revealing inclusion of cyclotomic subfields and providing explicit examples under certain congruence conditions.

Contribution

It establishes the inclusion of cyclotomic subfields in coefficient fields of newforms under specific congruence assumptions and provides explicit illustrative examples.

Findings

The maximal real subfield of the $ ext{l}$-th cyclotomic field is contained in the coefficient field of $f$.

Explicit examples demonstrate the theoretical results.

Splitting behavior of primes in coefficient fields is characterized under congruence conditions.

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.