Is the following analog of the cyclotomic polynomial irreducible in $\mathbb{Q}{[x]}$? $$\Phi_n(q;\chi) = \prod_{\gcd(k,n)=1}(q-\chi_n(k)\sqrt{\zeta_{n}}^k) \quad 1\leq k \leq n$$
Where $\chi_: \left(\mathbb{Z}/n\mathbb{Z}\right)^{\times}\rightarrow \mathbb{C}$ is a Dirichlet character, and $\zeta_n$ is the $n$th root of unity
For example:
$$\Phi_4(q;\chi) = q^2-\sqrt{2}q+1$$ I tried computing the first few by hand but couldn't see a pattern immediately. Also if this is already used/known somewhere than I would be very interested in a reference. Thanks
Edit:
I guess it would also be interesting to know if
$$\prod_{\gcd(k,nd)=1}(q-\chi_d(k)\zeta_{d}^k) \quad 1\leq k \leq n \,\,\,\text{and} \,\,\,d\leq n $$
(where $d$ is the conductor of $\chi$) is irreducible in $\mathbb{Q}{[x]}$?