A note on Maass forms of icosahedral type

TL;DR

This paper explores properties of Maass forms of icosahedral type, focusing on their Fourier coefficients and automorphy implications, without establishing a Galois representation but deriving related consequences.

Contribution

It extends the understanding of icosahedral Maass forms by analyzing their Fourier coefficients and automorphy aspects using Ramakrishnan's ideas, despite not proving Galois representation existence.

Findings

Weak automorphy of symmetric powers deduced

Implications for Fourier coefficient distribution

Connections to Chebotarev density theorem

Abstract

Using ideas of Ramakrishnan, we consider the icosahedral analogue of the theorems of Sarnak and Brumley on Hecke-Maass newforms with Fourier coefficients in a quadratic order. Although we are unable to conclude the existence of an associated Galois representation in this case, we show that one can deduce some implications of such an association, including weak automorphy of all symmetric powers and the value distribution of Fourier coefficients predicted by the Chebotarev density theorem.