# Classifications of Hecke Fields and Galois Images of Weight One Exotic Newforms

## TL;DR

This paper classifies the Hecke fields and Galois image structures of weight one exotic newforms, providing explicit descriptions based on the nebentypus and group type, advancing understanding of their arithmetic properties.

## Contribution

It offers a complete classification of Hecke fields and Galois images for weight one newforms of specific types, linking these to the nebentypus and group structure.

## Key findings

- Hecke fields are explicitly described in terms of nebentypus order.
- Galois image classifications are complete for each form type.
- Results enhance understanding of the arithmetic of weight one newforms.

## Abstract

We determine the Hecke fields associated with weight one newforms of $A_4$-, $S_4$-, and $A_5$-type, expressed in terms of the order of its nebentypus. Furthermore, for each type, we provide a complete classification of the images of the corresponding Galois representations.