9.Fractions of Powers (open)

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

For positive integers $k$ and $n$, define the function \(f_k(n)=\sum_{i=1}^n\lbrace\frac{i^k}{n}\rbrace\)

For example, $f_5(10)=4.5$ and $f_7(1234)=616.5$. Let \(S(N)=\sum_{\substack{k=1 \\ k \text { odd }}}^N \sum_{n=1}^N f_k(n)\)

You are given that $S(10)=100.5$ and $S\left(10^3\right)=123687804$.

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