Let $d \geq 2$ be an integer and $\xi=\exp(\frac{2\pi i}{d})$. I am trying to compute the determinant of the matrix $$ (\xi^{ij}-1)_{1 \leq i, j \leq d-1}. $$ Let me call it $\Delta(d)$. For small values of $d$ I get:

    $\Delta(2)=-2$
    $\Delta(3)=-3\sqrt{3}i$
    $\Delta(4)=-16i$

But I can't seem to find a general formula. How can I do this?