Twist-minimal trace formulas and the Selberg eigenvalue conjecture

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 [DM06], of conductor 1951. Second, we verify the Selberg eigenvalue conjecture for groups of small level, improving on a result of Huxley [Hux85] from 1985.