Modular Cubes, Part 1

For a positive number $n$ , define $S (n)$ as the sum of the integers $x$ , for which $1 < x < n$ and
$x^{3} \equiv 1 mod n$ .

When $n = 91$ , there are $8$ possible values for $x$ , namely: $9, 16, 22, 29, 53, 74, 79, 81$ .
Thus, $S (91) = 9 + 16 + 22 + 29 + 53 + 74 + 79 + 81 = 363$ .

Find $S (13082761331670030)$ .

Write-up coming later

The complete problem is available here. An approach, code, and answer will be added later.