Let $X$ be a chi squared variable with $121$ degrees of freedom. So the density $f_X$ of $X$ is defined by
$ f_X(x)=\frac{\big(\frac{x}{2}\big)^{\frac{121}{2}-1}}{\Gamma(\frac{121}{2})}{{e}^{-\frac{x}{2}}} $
I would like to compute $P(X>126)$ with an accuracy of $10^{-2}$. I know that a standard method is to approximate the distribution of $X$ by a normal distribution ( ${\cal N}(121,\sqrt{242})$ here), but I do not know of any control on the error made in this approximation. In theory this is just a problem of computing a definite integral with a good enough precision, but it seems to exceed the capacity of my formal calculator (indeed, the value $\Gamma(\frac{121}{2})$ is very large, its integer part has 81 digits.)
Is there a rigorous (and working!) method to solve this ?