Simple question: I was trying to find a formula for the Chern character $ch(S^mE)$ in terms of $ch(E)$ but couldn't find a reference too easily. It can be worked out using symmetric functions, generating functions, etc., but I didn't want to slog through, just to make a mistake in the end. If you have a reference or can reproduce the formula, I'd be much obliged!
P.S. The rank of $E$ is $2$, in case that makes things easier.
Edit: It is rather simple for rank $2$ it seems using the splitting principle, but I thought I'd keep this question up for posterity, given that the higher rank case is certainly a bit trickier.