Possible connection between a generalized Maeda's conjecture and local types

TL;DR

This paper explores a potential link between a generalized Maeda's conjecture and local types, supported by computational evidence and extended formulas, with implications for Galois groups and totally real fields.

Contribution

It proposes a new conjectural relationship between local types and Galois orbits, extending Maeda's conjecture to broader settings including non-trivial Nebentypus.

Findings

Computational evidence supports the conjectured relation.

Derived a formula for non-CM Galois orbits with trivial Nebentypus.

Numerical data suggests possible generalizations to totally real fields.

Abstract

Here we follow on the proposed generalization of Maeda's conjecture made in [2]. We report on computations that suggest a relation between the number of local types and the number of non-CM newform Galois orbits. We extend the conjecture into spaces with non-trivial Nebentypus and provide a formula for the number of non-CM orbits for all levels and trivial Nebentypus. We also provide some numerical evidence towards further generalizations of this conjecture to totally real fields as well as further strengthening of it by proposing a structure for the corresponding Galois groups.