Slopes of the U_7 Operator Acting on a Space of Overconvergent Modular Forms

TL;DR

This paper explicitly computes the slopes of the U_7 operator on overconvergent modular forms with specific characters, leading to a complete understanding of slopes for classical Hecke newforms of certain weights and characters.

Contribution

It provides explicit calculations of U_7 slopes on overconvergent modular forms with a specific Dirichlet character, extending to classical Hecke newforms.

Findings

Explicit slopes of U_7 on overconvergent forms are computed.

Results apply to all classical Hecke newforms with the given weight and character.

The work connects explicit calculations with Coleman and Cohen-Oesterlé's theoretical results.

Abstract

Let \chi\ be the primitive Dirichlet character of conductor 49 defined by \chi(3)=\zeta, for \zeta\ a primitive 42nd root of unity. We explicitly compute the slopes of the U_7 operator acting on the space of overconvergent modular forms on X_1(49) with weight k and character either \chi^{7k-6} or \chi^{8-7k}, depending on the embedding of Q(\zeta) into C_7. By applying results of Coleman, and of Cohen-Oesterl\'e, we are then able to conclude the slopes of U_7 acting on all classical Hecke newforms of the same weight and character.