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:

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

**Notations:**

- \(i = \sqrt{-1}\) denotes the imaginary unit.
- \(e \approx 2.71828\) denotes the Euler's number.

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