Newforms and spectral multiplicity for Γ₀ (9)

Abstract

The goal of this paper was to explain certain experimentally observed properties of the (cuspidal) spectrum and its associated automorphic forms (Maass waveforms) on the congruence subgroup Γ0(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 Γ0(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 Γ0(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 Γ0 (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.

Noncongruence subgroups and Maass waveforms

“…[1]. It is also consistent with the notion of genuinely new Maass waveforms introduced in [66]. By using the description of the setC(G , G) below it is also easy to verify that the definition of old space agrees with the definition above for congruence subgroups.…”

Computations of Eigenvalues and Resonances on Perturbed Hyperbolic Surfaces with Cusps

“…A subset of these eigenvalues are eigenvalues for the groups Γ 0 (5), Γ 0 (6), and Γ 3 . These are in good agreement with those computed in [22] and [33] C arccosh(2),0 C arccosh(3),0 C arccosh(9),0 C 2 arccosh( 3 2 ),1/2 C 2 arccosh(2),1/2 and and C arccosh(5),0 C 2 arccosh(3),1/2 t µ(t) t µ(t) t µ(t) t µ(t) t µ(t)…”

Spectral correspondences for Maass waveforms on quaternion groups

“…Thus, unless there are some non-trivial "oldforms" showing up, our claim is that there is no order in the case N = 4pu 2 that gives the same correspondence as in Theorem 8.1, but we have no proof of it. There could for example be a phenomenon analogous to the one for Γ 0 (9), where all newforms actually are forms, or twists of forms, corresponding to groups strictly containing Γ 0 (9) [29]. Hence, we definitely do not exclude the possibility of finding a correspondence with an adjusted notion of newforms in the case N = 4pu 2 .…”