Hypergeometric 3f 2(1/4) Evaluations over Finite Fields and Hecke Eigenforms

Let H denote the hypergeometric 3F2 function over Fp whose three numerator parameters are quadratic characters and whose two denominator parameters are trivial characters. In 1992, Koike posed the problem of evaluating H at the argument 1/4. This problem was solved by Ono in 1998. Ten years later, Evans and Greene extended Ono’s result by evaluating an infinite family of 3F2(1/4) over Fq in terms of Jacobi sums. Here we present five new 3F2(1/4) over Fq (involving characters of orders 3, 4, 6, and 8) which are conjecturally evaluable in terms of eigenvalues for Hecke eigenforms of weights 2 and 3. There is ample numerical evidence for these evaluations. We motivate our conjectures by proving a connection between 3F2(1/4) and twisted sums of traces of the third symmetric power of twisted Kloosterman sheaves.