LMS Journal of Computation and Mathematics

Research Article

Slopes of the U7 operator acting on a space of overconvergent modular forms

L. J. P. Kilforda1 and Ken McMurdya2

a1 School of Mathematics, University of Bristol, University Walk, Bristol, BS8 1TW, United Kingdom (email: l.kilford@gmail.com)

a2 Department of Mathematics (TAS), Ramapo College of New Jersey, 505 Ramapo Valley Rd, Mahwah NJ 07430, USA (email: kmcmurdy@ramapo.edu)

Abstract

Let χ be the primitive Dirichlet character of conductor 49 defined by χ(3)=ζ for ζ a primitive 42nd root of unity. We explicitly compute the slopes of the U7 operator acting on the space of overconvergent modular forms on X1(49) with weight k and character χ7k−6 or χ8−7k, depending on the embedding of xs211A(ζ) into xs21027. By applying results of Coleman and of Cohen and Oesterlé, we are then able to deduce the slopes of U7 acting on all classical Hecke newforms of the same weight and character.

(Received June 24 2011)

(Revised January 28 2012)

(Online publication May 2012)

2010 Mathematics Subject Classification

  • 11F11;
  • 11F33 (primary);
  • 14G35;
  • 14G22 (secondary)