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GENERATING FUNCTIONS FOR HECKE OPERATORS

HALA AL HAJJ SHEHADEH

Department of Mathematics, Courant Institute of Mathematical Sciences, 251 Mercer Street, New York, NY 10012, USA

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SAMAR JAAFAR

Mathematics and Physics Department, Qatar University, P. O. Box 2173, Doha, Qatar

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, and

KAMAL KHURI-MAKDISI

Center for Advanced Mathematical Sciences, American University of Beirut, Bliss Street, Beirut, Lebanon

Corresponding author.

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https://doi.org/10.1142/S1793042109001979Cited by:1 (Source: Crossref)

Abstract

Fix a prime N, and consider the action of the Hecke operator T_N on the space of modular forms of full level and varying weight κ. The coefficients of the matrix of T_N with respect to the basis {E₄ⁱ E₆^j | 4i + 6j = κ} for can be combined for varying κ 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.

Keywords:

AMSC: 11F32, 13D40, 11F11

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