This “(-1 - i)” you see is actually a root of unity, at least if you’re looking at my solution.
The person above is correct that the transformation matrix will look like this:
\begin{pmatrix} 1 & 1 & 1 \\ 1 & \omega & \omega^2 \\ 1 & \omega^2 & \omega \end{pmatrix}.
Here \omega^3 = 1, \omega \ne 1. The problem? No such element exists modulo 10^9 + 7. The only cube root of unity is 1. We don’t want to go to complex numbers either, the values will get really huge and/or we have to awkwardly take many transforms and inverses.
So what’s the idea here? We will use some finite field theory to extend the field of numbers mod 10^9 + 7 to have such an \omega. This is somewhat awkward to explain but basically, we can use the field \mathbb{F}_{10^9 + 7}[x] / (x^2 + x + 1) (polynomials modulo x^2 + x + 1). Each element has two “coordinates”, both coordinates are modulo 10^9 + 7. Addition is pointwise, multiplication has some slightly awkward rule.