Generating functions for Hecke operators
Hala Hajj Shehadeh, Samar Jaafar, Kamal Khuri-Makdisi
Abstract
Fix a prime N, and consider the action of the Hecke operator TN on the space Mk(SL(2,Z)) of modular forms of full level and varying weight k. The coefficients of the matrix of TN with respect to the basis E4i E6j | 4i + 6j = k for Mk(SL(2,Z)) can be combined for varying k 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.
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