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 $0 \le \{x\} \lt 1$.
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.