Number of grid 11x11

Let $n \ge 2$ be an odd positive integer and let $p$ be an odd prime. Consider an $n \times n$ square grid in which each unit square is filled with an integer from the set $\{1, 2, \ldots, p-1\}$. Such a grid is called good if it satisfies both of the following conditions: • The product of the $4$ numbers in every $2 \times 2$ square of the grid is congruent to $1$ modulo $p$. • The product of all $n^2$ numbers in the grid is congruent to $-1$ modulo $p$. Two grids are considered different if there is a unit square, in some row $i$ and column $j$, that contains different numbers in the two grids. Let $N$ be the number of good grids for $n = 11$ and $p = 7$. Find the remainder when $N$ is divided by $1000$.