Newforms and Spectral Multiplicities for $\Gamma_0(9)$

TL;DR

This paper analyzes the spectral properties of the congruence subgroup a09, revealing multiplicities in the spectrum, the absence of genuinely new eigenvalues, and the existence of eigenvalues with eigenspaces of dimension at least two, related by character twists.

Contribution

It provides a detailed spectral decomposition of a09, proving the genuinely new spectrum is empty and constructing explicit examples of eigenforms related by character twists.

Findings

Spectral multiplicities are present in the new part of a09.

No genuinely new eigenvalues exist in the spectrum.

Existence of eigenvalues with eigenspaces of dimension at least two, related by character twists.

Abstract

The goal of this paper is to explain certain experimentally observed properties of the (cuspidal) spectrum and its associated automorphic forms (Maass waveforms) on the congruence subgroup $\Gamma_{0}(9)$. The first property is that the spectrum possesses multiplicities in the so-called new part, where it was previously believed to be simple. The second property is that the spectrum does not contain any "genuinely new" eigenvalues, in the sense that all eigenvalues of $\Gamma_{0}(9)$ appear in the spectrum of some congruence subgroup of lower level. The main theorem in this paper gives a precise decomposition of the spectrum of $\Gamma_{0}(9)$ and in particular we show that the genuinely new part is empty. We also prove that there exist an infinite number of eigenvalues of $\Gamma_{0}(9)$ where the corresponding eigenspace is of dimension at least two and has a basis of pairs of Hecke-Maass newforms which are related to each other by a character twist. These forms are non-holomorphic analogues of modular forms with inner twists and also provide explicit (affirmative) examples of a conjecture stating that if the Hecke eigenvalues of two "generic" Maass newforms coincide on a set of primes of density 1/2 then they have to be related by a character twist.