GENERATING FUNCTIONS FOR HECKE OPERATORS

Abstract

Fix a prime N, and consider the action of the Hecke operator TN on the space [Formula: see text] of modular forms of full level and varying weight κ. The coefficients of the matrix of TN with respect to the basis {E4i E6j | 4i + 6j = κ} for [Formula: see text] can be combined for varying κ into a generating function FN. We show that this generating function is a rational function for all N, and present a systematic method for computing FN. We carry out the computations for N = 2, 3, 5, and indicate and discuss generalizations to spaces of modular forms of arbitrary level.