Consider $a\in \mathbb R^n$ a random variable entries $a \sim \mathcal N(0,1)$ (Gaussian distribution with mean 1), i.i.d. elementwise. Then $$ z = (a^Tu)^2 a^TCa $$ is another random variable. I want to determine its mean, which I can then use to approximate $u^TCu$ via some concentration inequality.
Any ideas on how to find $\mathbb E[z]$ the expectation of $z$?