What would be the distribution of the following ratio

$z = \frac{x_{1}}{|x_{1}|^2 + |x_{2}|^2 + ... + |x_{M}|^2}$

where $x \sim \mathcal{CN}(0,a)$ and $a > 1$. As can be seen, the denominator follows a Chi-square distribution with $2M$ degrees of freedom.

Remark 1: I've run some simulations in Matlab and it seems that the resulting distribution is Gaussian, or at least, it has a bell-shaped histogram.