Given following density function $$f(x,\theta) = \frac{1}{3}\theta^{-4} x^7 \exp{-x^2/\theta},$$ I'm calculating the method of moments estimator and its asymptotic normality result. For the method of moments estimator I found $$\hat{\theta}^{\text{MoM}}_n = \left(\frac{32}{35}\right)^2 \frac{\bar{x}}{\pi}.$$ I'm however not entirely sure this is the right answer. Also I have no clue on how to get the asymptotic normality result for this estimator. I'm assuming it has to do with the central limit theorem.
$\textbf{EDIT: }$ Using the CLT I have found this: $$\hat{\theta}^{\text{MoM}}_n \sim \mathcal{N}\left(\frac{32}{35}\sqrt{\theta}, \frac{1}{n}\left( \frac{32^2}{35^2} \frac{5!}{6\sqrt{\pi}} \theta - \theta \sqrt{\pi} \right) \right)$$ but this seems very very messy, making me believe that it is incorrect...
There is a hint given: $$E(X^r) = \frac{1}{6}\Gamma(4 + r/2)θ^{r/2}$$