Complexity! (6)

Number Theory Level 3

If $$\displaystyle \alpha = e^{\frac{i2\pi}{7}}$$ and $$f(x) = 1 + \displaystyle \sum_{k=1}^{6}{a_{k}x^{k}} + \sum_{k=8}^{13}{a_{k}x^{k}} + \sum_{k=15}^{20}{a_{k}x^{k}}$$, then find the value of the expression below:

$f(x) + f(\alpha x) + f({\alpha}^{2} x) + f({\alpha}^{3} x) + f({\alpha}^{4} x) + f({\alpha}^{5} x) + f({\alpha}^{6} x)$

Notation: