SU(2)-group elements can be written in terms of Euler-angles $\alpha,\beta,\gamma$ (using the convention from this wikipedia article, i.e. z-y-z convention). Now, it is stated that the characters of the spin-j representation of an arbitrary group element $g=g(\alpha,\beta,\gamma)\in\mathrm{SU}(2)$ are given by
$$\chi^{j}(g)=\frac{\mathrm{sin}((2j+1)\beta/2)}{\mathrm{sin}(\beta/2)}$$
Now, I would like to now if it is possible to write down the expression for $\chi^{j}(g^{2})$, or maybe more general, $\chi^{j}(g^{n})$.
My attempt: I tried to calculate $g^{2}$ explicitely, in order to determine the Euler angles of $g^{2}$, but this becomes quite messy...