I would like to compute the Fourier series given as
$f(2\pi k/n)= \sum\limits_{\substack{k=-N}}^{N}{t_{k} e^{(i2 \pi k^2/n)}} $
I have in my work that $t_{k}=t_{-k}$, then we are in a symmetric case, then the series can be expressed as :
$f(2\pi k/n)= 1/2+ 2 \sum\limits_{\substack{k=1}}^{N}{t_{k} \cos(2 \pi k^2/n)}$.
Now the question is what could be a possible approach to find the sum of this series $\sum\limits_{\substack{k=1}}^{N}{\cos(2 \pi k^2/n)} $?
I don't take into account $t_{k}$ because they have all the same value in my work.