Residue calculation for $z^{-2}\sin(z^{-2})$

Question

I am trying to calculate the residue of $f(z) = \frac{1}{z^2}\sin\left(\frac{1}{z^2}\right)$ at $z=0$. Looking at the Taylor expansion of $f$ around $0$, as $c_{-n}$ (the coefficients of the powers of $z$) is non-zero for infinitely many $n$, $z=0$ must be a essential singularity of the function. However, as the expansion does not contain a term proportional to $\frac{1}{z}$, does this just mean $\text{Res}_{z=0}f(z) = 0$ by the definition of the Residue? This doesn't feel correct.