Find the maximum of $g(ξ)=(e^{πξi}-1)\sum\limits_{k=0}^na_ke^{2πξik}$

Question

Let $\xi\in\mathbb R$, $n\in\mathbb N$, $\{a_k\}_{k=0}^n\in\mathbb R$. I wonder if it is possible to calculate or estimate the maximum of $|g(\xi)|$, where $$g(\xi)=(e^{\pi \xi i}-1)\sum_{k=0}^n a_k e^{2\pi \xi i k}\, .$$ My attempt: $$\max_\xi|g(\xi)|\leq \max_\xi\left|e^{\pi \xi i}-1\right|\, \max_\xi\left|\sum_{k=0}^n a_k e^{2\pi \xi i k}\right|\leq \max_\xi\left|e^{\pi \xi i}-1\right|\, \sum_{k=0}^n \left|a_k \right|\, .$$ Can we do better?