A $p$-adic ($p\equiv 3\!\!\pmod 4$) depth-$5$ supercongruence for Gaussian $p$-th power sums over a square

Abstract: Let $p$ be an odd prime. Define the Gaussian power sum [ \G_n(p)=\sum_{a=1}^{p-1}\sum_{b=1}^{p-1}(a+b\ii)^n\in\ZZ[\ii]. ] We determine $\G_p(p)$ modulo high powers of $p$: if $p\equiv 1\pmod 4$ then $$\G_p(p)\equiv p<sup>2(1+\ii)\pmod{p<sup>3},$$ while for $p\equiv 3\pmod 4, p\ge 7$ we prove the supercongruence [ \G_p(p)\equiv -\frac{p^5}{12}(p-1)^2(p-2)\,B_{p-3}\,(1-\ii)\pmod{p^6}, ] where $B_m$ denotes the $m$-th Bernoulli number. We also formulate several conjectures suggested by extensive computations.