# Calculating eigenvalues of G Parity

## Homework Statement

I need to find the eigenvalues of the pion triplet under G parity

## Homework Equations

$$G\mid\psi\rangle = CR_2\mid\psi\rangle$$

## The Attempt at a Solution

OK so visually this problem is pretty simple, rotation about the 2 axis takes a pi+ to a pi- and then charge takes it back to a pi+ (to a sign). The problem is when I actually try to do the calculation, the rotation gives me $\pi^\pm\rightarrow\mp\pi^0 \mathrm{\ and\ }\pi^0\rightarrow\pi^++\pi^-$, which obviously are eigenstates of C but the pions are not eigenstates of G. My rotation matrix is the standard:
R2 = [(010),(-101),(0-10)] (ignoring the constants).
It is easy to construct a matrix that gives the desired transformations [(001),(010),(100)], but it would not be traceless and thus isn't a rotation matrix.

I found plenty of resources that just say that the fact that G parity works is obvious (which it is) but none that actually show how to do the calculation. Any help would be greatly appreciated..