Let $X\sim N(0,1)$ and $Y=X²-1$. Show that $X$ and $Y$ are uncorrelated but not independent.

Question

I've got some problems with the following task:

Let $X\sim N(0,1)$ and $Y=X²-1$. Show that $X$ and $Y$ are uncorrelated but not independent. I showed that $X$ and $Y$ are uncorrelated but I'm not sure how to proof the dependence of $X$ and $Y$. Can anybody give me some hints? I calculated the density function of $Y$ but could not see, how this could help.