Let f be a weight 2 normalized newform on the congruence subgroup Γ0(N) with integral Fourier coefficients. There are two important numerical invariants attached to f : its congruence number and its modular degree. By definition, the congruence number of f is the largest integer Df such that there exists a weight 2 cusp form on Γ0(N), with integral coefficients, which is orthogonal to f with re...

Until recently, no Salem numbers were known of trace below −1. In this paper we provide several examples of trace −2, including an explicit infinite family. We establish that the minimal degree for a Salem number of trace −2 is 20, and exhibit all Salem numbers of degree 20 and trace −2. Indeed there are just two examples. We also settle the closely-related question of the minimal degree d of a...

To explain the determinants of negative behavior toward deviants (e.g., punishment), this article examines how people evaluate others on the basis of two types of standards: minimal and maximal. Minimal standards focus on an absolute cutoff point for appropriate behavior; accordingly, the evaluation of others varies dichotomously between acceptable or unacceptable. Maximal standards focus on th...