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Eta-products which are simultaneous eigenforms of Hecke operators

Published online by Cambridge University Press:  18 May 2009

Anthony J. F. Biagioli
Anthony J. F. Biagioli
Affiliation:
University Of Central ArkansasConway, Arkansas, USA Pennsylvania State University, Berks Campus Reading, Pennsylvania, USA
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The Dedekind eta-function is defined for any τ in the upper half-plane by

where x = exp(2πiτ) and x1/24 = exp(2πiτ/24). By an eta-product we shall mean a function

where N ≥ 1 and eachrδ ∈ ℤ. In addition, we shall always assume that is an integer. Using the Legendre-Jacobi symbol (—), we define a Dirichlet character ∈ by

when a is odd. If p is a prime for which ∈(p) ≠ 0and if F is a function with a Fourier series

then we define a Hecke operator Tp by

where

and

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 35 , Issue 3 , September 1993 , pp. 307 - 323
Copyright
Copyright © Glasgow Mathematical Journal Trust 1993

References

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