Abstract
We derive a fully explicit version of the Selberg trace formula for twist-minimal Maass forms of weight 0 and arbitrary conductor and nebentypus character, and apply it to prove two theorems. First, conditional on Artin's conjecture, we classify the even 2-dimensional Artin representations of small conductor; in particular, we show that the even icosahedral representation of smallest conductor is the one found by Doud and Moore, of conductor 1951. Second, we verify the Selberg eigenvalue conjecture for groups of small level, improving on a result of Huxley from 1985.
Highlights
In [6], the first and third authors derived a fully explicit version of the Selberg trace formula for cuspidal Maass newforms of squarefree conductor, and applied it to obtain partial results toward the Selberg eigenvalue conjecture and the classification of 2-dimensional Artin representations of small conductor
We remove the restriction to squarefree conductor, with the following applications: Theorem 1.1
As we pointed out in [6], in the case of squarefree level the Selberg trace formula becomes substantially cleaner if one sieves down to newforms, and that helps in numerical applications by thinning out the spectrum
Summary
In [6], the first and third authors derived a fully explicit version of the Selberg trace formula for cuspidal Maass newforms of squarefree conductor, and applied it to obtain partial results toward the Selberg eigenvalue conjecture and the classification of 2-dimensional Artin representations of small conductor. As we pointed out in [6], in the case of squarefree level the Selberg trace formula becomes substantially cleaner if one sieves down to newforms, and that helps in numerical applications by thinning out the spectrum. The reduction to twist-minimal spaces yields a substantial improvement in our numerical results by essentially halving the spectrum in the critical case of forms of prime level N and conductor N 2 (see (1.3)). This partially explains why our result for Γ(N ) is within a factor of 4 of that for Γ1(N ), despite the conductors being much larger.
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