Splitting behaviour of primes in coefficient fields of newforms in terms of level and nebentypus

Determine how the level N and the nebentypus character ε of a newform f ∈ S_k(Γ_0(N), ε) constrain the splitting behaviour of rational primes in the coefficient field Q_f = Q({a_n(f)}_{n≥1}), that is, characterize the splitting, inertia, and ramification of primes of Q in Q_f as functions of N and ε.

Background

The paper studies the arithmetic of the coefficient field Q_f of a newform f, focusing on how local properties at primes dividing the level influence global splitting in Q_f. While specific results are proved under hypotheses involving congruences and loss of ramification at a prime p, the broader objective is to understand the general relationship between the level N, the nebentypus ε, and the splitting of primes in Q_f.

The authors provide theorems showing that under certain congruence and conductor-lowering conditions, Q_f contains cyclotomic subfields such as Q(ζ_ℓ) or its maximal real subfield, leading to strong ramification constraints at ℓ. However, the overarching question seeks a comprehensive description that does not rely on these specific hypotheses.

References

Although modular forms have been extensively studied, there are still many open questions about the field of coefficients Q_f. Here we are particularly interested in the following. What can be said about the splitting behaviour of primes in Q_f in terms of the level N and the nebentypus ε?

— Congruences and ramified primes in fields of coefficients of newforms  (2603.29468 - Freitas et al., 31 Mar 2026) in Introduction, Question (page 1)