Let $G=C_n$ be the cyclic group and $f_n$ the characteristic polynomial of its character table (over $\mathbb{C}$) in the ordering so that the character table is given by the discrete Fourier transform matrix (https://en.wikipedia.org/wiki/DFT_matrix) without the factor.

Question 1: For which $n$ is $f_n$ irreducible (over the smallest field extension of $\mathbb{Q}$ containing its coefficients)?

I did input this into GAP using the command IrrDixonSchneider to obtain the character table (I hope this is does not change much with respect to irreducibility) and tested for irreducibility with GAP. GAP seems to use another numbering for the character table, so lets call $g_n$ the characteristic polynomial according to GAP (maybe someone can clarify the ordering which GAP uses to get the character table of the cyclic group. I try to see what it is but Im not sure at the moment).

For $n \leq 23$ it was true that the $n$ such that $g_n$ is irreducible coincides with the sequence https://oeis.org/A280862 , which are those $n$ such that $a_n \psi_n = \varphi_n$, where $\varphi_n$ is the Euler phi function (https://oeis.org/A000010), $\psi_n$ is the reduced totient function (https://oeis.org/A002322) and $a_n$ is the greatest common divisor of all $(d-1)$'s, where the $d$'s are the positive divisors of $n$ (https://oeis.org/A258409).

Question 2: Is this true?

Question 3: Does being irreducible depend on the ordering used to obtain the character table here (or even for a general group)?

notin natural bijection, so there is no way to view the character table as naturally the matrix of an endomorphism of a vector space. $\endgroup$ – Joshua Mundinger yesterday