As you know, we can write a $2\pi$ periodic function as a complex Fourier series $$ f(x) = \sum_{n=-\infty}^{+\infty} c_n e^{inx}, \quad f_n = e^{inx} $$
Where $c_n$ are the complex Fourier coefficients, $$ c_n = \langle f_n | f \rangle = \int_{-\pi}^{\pi} f(x) e^{-inx} dx $$
However, in an exam exercice, I was asked to find the Fourier coefficients such that $$ f(x) = \frac{1}{\sqrt{2\pi}} \sum_{n=-\infty}^{+\infty} c_n e^{inx} $$
FYI, the function was, $$ f: [-\pi, \pi] \rightarrow \mathbb{R}, \quad f(x) = x^3 - \pi^2 x $$
In that case, does the expression of $c_n$ changes ? If so, how do we compute the $c_n$ for the expression above ?