GENERATING FUNCTIONS FOR HECKE OPERATORS

Fix a prime N, and consider the action of the Hecke operator T-N on the space M-kappa(SL(2, Z)) of modular forms of full level and varying weight kappa. The coefficients of the matrix of TN with respect to the basis {(E4E6j)-E-i vertical bar 4i + 6j = kappa} for M-kappa(SL(2, Z)) can be combined for varying. into a generating function F-N. We show that this generating function is a rational function for all N, and present a systematic method for computing F-N. We carry out the computations for N = 2, 3, 5, and indicate and discuss generalizations to spaces of modular forms of arbitrary level.