Fourier series of $\csc^2(a\cdot z)$

Question

Looking at the Fourier series of the Weierstrass $\wp$ function, I have a term going like $\csc^2(az)$. A straightforward method to get its Fourier series is writing it like $$ \csc^2(az)=\frac{4}{2-e^{2iaz}-e^{-2iaz}}. $$ Mathematica returns the series $$ \csc^2(az)\sim -\sum_{n=1}^\infty 4n e^{2inaz}. $$ Could one make sense of this result or does it exist a known formula I am not aware of?