# Number Theory Seminar

Title: New congruences for eigenforms of level 1.

Abstract: In this talk, we will state recently discovered congruences satisfied by Hecke eigenforms of level 1. These congruences extend previous results of Hatada (1981), and prove a conjecture of Coleman-Stein (2003). They can be seen as further evidence for a conjecture, formulated jointly with Kiming and Wiese (2016), on the finiteness of congruence classes of eigenforms of fixed level modulo prime powers. The proof uses Merel's description of the action of Hecke operators on modular symbols. We will give a brief sketch of Merel's modular symbol formalism in level 1 and describe the algorithm used to obtain these new congruences.