Covariance between a standard normal and its square

Let $Z$ be standard normal, and define $X=Z$ and $Y=Z^2$. What is $\operatorname{Cov}(X,Y)$?

$0$. $\operatorname{Cov}(Z,Z^2)=\mathbb{E}[Z^3]-\mathbb{E}[Z]\,\mathbb{E}[Z^2]$, and both $\mathbb{E}[Z^3]$ and $\mathbb{E}[Z]$ vanish by symmetry of the normal. This is a classic example of uncorrelated yet dependent variables.