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 (J. Number Theory 118 (2006) 62–70) of conductor 1951. Second, we verify the Selberg eigenvalue conjecture for groups of small level, improving on a result of Huxley (Elementary and analytic theory of numbers, Banach Center Publications 17 (PWN, Warsaw, 1985) 217–306) from 1985.

中文翻译：

---

扭曲最小跟踪公式和Selberg特征值猜想

我们为权重为0的扭曲最小马斯形式以及任意导体和nebentypus特征导出了Selberg迹线公式的完全显式版本，并将其应用于证明两个定理。首先，根据阿丁的猜想，我们对小导体的偶数二维阿丁表示进行分类。特别是，我们证明了最小导体的偶数二十面体表示是由导体1951年的Doud和Moore（J. Number Theory 118（2006）62–70）发现的。其次，我们验证了小群的Selberg特征值猜想，从1985年起，对赫x黎（数字的基本和解析理论，巴纳赫中心出版物17（PWN，华沙，1985）217–306）的结果进行了改进。