Hypergeometric $\mathbf{{}_3{\it F}_2(1/4)}$ evaluations over finite fields and Hecke eigenforms

Let $H$ denote the hypergeometric ${}_3F_2$ function over $\mathbb{F}_p$ 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 ${}_3F_2(1/4)$ over $\mathbb{F}_q$ in terms of Jacobi sums. Here we present five new ${}_3F_2(1/4)$ over $\mathbb{F}_q$ (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 ${}_3F_2(1/4)$ and twisted sums of traces of the third symmetric power of twisted Kloosterman sheaves.