Exotic newforms constructed from a linear combination of eta quotients

TL;DR

This paper constructs exotic newforms from eta quotients, identifies their associated Galois extensions and representations, and analyzes prime splitting, advancing understanding of their arithmetic properties.

Contribution

It explicitly finds Galois extensions and representations for newforms built from eta quotients, confirming the Deligne-Serre correspondence in these cases.

Findings

Identified Galois fields and representations for specific newforms

Analyzed prime splitting behavior in these Galois extensions

Classified all such newforms from eta quotients with given q-expansions

Abstract

K{\"o}hler, in [1], presented a weight 1 newform on $\Gamma_0(576)$ constructed from a linear combination of weight 1 eta quotients and asked, ``What would be a suitable $L$ and representation $\rho$ such that Deligne\text{-}Serre correspondence holds?" In this paper, we find the Galois field extension $L$ and representation $\rho$ such that the Deligne\text{-}Serre correspondence holds for this newform, and also study the splitting of primes in $L$ using the coefficients $a(p)$ of the newform. We also discuss an exotic newform on $\Gamma_0(1080)$ constructed from a linear combination of weight 1 eta quotients, find the corresponding Galois extension and representation, and study the splitting of primes in this extension. Furthermore, we find all such newforms that can be constructed from a linear combination of weight 1 eta quotients listed in [2] with $q$-expansion of the form $q+\sum_{k=2}^{\infty}a(k)q^k$.