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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

A simplified trace formula for Hecke operators for $ \Gamma\sb 0(N)$


Author: Shepley L. Ross
Journal: Trans. Amer. Math. Soc. 331 (1992), 425-447
MSC: Primary 11F11; Secondary 11F25, 11F72
MathSciNet review: 1053115
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Abstract: Let $ N$ and $ n$ be relatively prime positive integers, let $ \chi $ be a Dirichlet character modulo $ N$, and let $ k$ be a positive integer. Denote by $ {S_k}(N,\chi)$ the space of cusp forms on $ {\Gamma _0}(N)$ of weight $ k$ and character $ \chi $, a space denoted simply $ {S_k}(N)$ when $ \chi $ is the trivial character. Beginning with Hijikata's formula for the trace of $ {T_n}$ acting on $ {S_k}(N,\chi)$, we develop a formula which essentially reduces the computation of this trace to looking up values in a table. From this formula we develop very simple formulas for (1) the dimension of $ {S_k}(N,\chi)$ and (2) the trace of $ {T_n}$ acting on $ {S_k}(N)$.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1992-1053115-1
PII: S 0002-9947(1992)1053115-1
Keywords: Cusp forms, Hecke operators, modular forms, trace formula
Article copyright: © Copyright 1992 American Mathematical Society