Sums of Power Sums

Let $f_{k} (n)$ be the sum of the $k$ ^th powers of the first $n$ positive integers.

For example, $f_{2} (10) = 1^{2} + 2^{2} + 3^{2} + 4^{2} + 5^{2} + 6^{2} + 7^{2} + 8^{2} + 9^{2} + 10^{2} = 385$ .

Let $S_{k} (n)$ be the sum of $f_{k} (i)$ for $1 \leq i \leq n$ . For example, $S_{4} (100) = 35375333830$ .

What is $\sum (S_{10000} (10^{12}) mod p)$ over all primes $p$ between $2 \cdot 10^{9}$ and $2 \cdot 10^{9} + 2000$ ?

Write-up coming later

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