exponentiation - Complex numbers, finding smallest exponent

$$z= \frac{\sqrt 3 -i}{1+ \sqrt3 i}$$
I need to find smallest exponent $n>2018$, such as $z^n$ will be a number with real part equal to $0$ and imaginary part of number will be negative.

I calculated that $z=-i$ but I don't know how to write a equation that covers all conditions.
It doesn't have to be an exact answer, I am asking more about how do I solve this kind of a problem effectively