The norm of the inverse Fourier transform

Question

Let $C_n$ be the cyclic group of order $n$ and consider the finite Fourier transform as a linear map

$$\mathcal{F}_n\colon \ell_1(C_n)\to \ell_\infty^n,$$

where $\ell_\infty^n$ is $\mathbb{C}^n$ with the max norm.

Is $\|\mathcal{F}_n^{-1}\|$ computed anywhere in the literature?

It is not hard to show that these numbers grow to infinity.

Answer 1

For $n=4$ a script showed a maximizer of $$\max_{x \in \Bbb{C}^4, \ \ \|x\|_\infty=1} \|Fx\|_1$$ is
$$x\approx ( 1, e^{-2.40101 i},e^{ 0.00085 i}, e^{ 0.74117 i}),\qquad \|F x\|_1 = \color{red}{8}$$