Fractions of Powers (noch nicht übersetzt)

Any positive real number $x$ can be decomposed into integer and fractional parts $\lfloor x \rfloor + \{x\}$, where $\lfloor x \rfloor$ (the floor function) is an integer, and $0\le \{x\} < 1$.

For positive integers $k$ and $n$, define the function \begin{align} f_k(n) = \sum_{i=1}^{n}\left\{ \frac{i^k}{n} \right\} \end{align} For example, $f_5(10)=4.5$ and $f_7(1234)=616.5$.

Let \begin{align} S(N) = \sum_{\substack{k=1 \\ k\text{ odd}}}^{N} \sum_{n=1}^{N} f_k(n) \end{align} You are given that $S(10)=100.5$ and $S(10^3)=123687804$.

Find $\lfloor S(33557799775533) \rfloor$. Give your answer modulo 977676779.