Generating functions for Hecke operators

Details Generating functions for Hecke operators Generating functions for Hecke operators

2006-10-31

arxiv.org/abs/math/0610962v1

International Journal of Number Theory 5 (2009), no. 1, 125-140

Mathematics

Number Theory

13 pages

Scientific paper

Fix a prime N, and consider the action of the Hecke operator T_N on the space M_k(SL(2,Z)) of modular forms of full level and varying weight k. The coefficients of the matrix of T_N with respect to the basis {E_4^i E_6^j | 4i + 6j = k} for M_k(SL(2,Z)) can be combined for varying k 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.

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