The value of $\sum_{n=1}^{\infty} \frac{{(-1)}^n}{n^6}$ using the Fourier Transformation

Question

How to compute the sum given in $(1)$, from the fourier transformation of $\pi x |x|-x^3$ over $(-\pi,\pi)$. $$ \sum_{n=1}^{\infty} \frac{{(-1)}^n}{n^6} \tag{1} $$

Suppose we know that $\sum_{n=1}^{\infty} \frac{1}{n^6}=\frac{\pi^6}{945}$.

Thanks for any help.