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International Mathematics Research Notices (2004) 2004:3233-3262 , doi:10.1155/S1073792804132522
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Copyright © 2004 Hindawi Publishing Corporation. All rights reserved.

Gaussian hypergeometric functions and traces of Hecke operators

Sharon Frechette, Ken Ono and Matthew Papanikolas

We establish a simple inductive formula for the trace Formula of the pth Hecke operator on the space Formula of newforms of level 8 and weight k in terms of the values of 3F2-hypergeometric functions over the finite field Fp. Using this formula when k = 6, we prove a conjecture of Koike relating Formula to the values 6F5(1)p and 4F3(1)p. Furthermore, we find new congruences between Formula and generalized Apéry numbers.


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