Let $z$ satisfying the equation $z^3=1$ be a generator of the cyclic group $\mathbb{Z}_3= \{ 1 , z,z^2 \}$. You are given that $\rho : \mathbb{Z}_3 \to GL(\mathbb{C}^2)$ defined by $$\rho(z) = \begin{bmatrix} -1 & 1 \\ -1 & 0 \end{bmatrix}$$ is a representation of the group $\mathbb{Z}_3$ in the vector space $\mathbb{C}^2$.
(a) Find an inner product on the complex vector space $\mathbb{C}^2$ which is $\rho$-invariant: $$\langle \rho(g)u,\rho(g)v \rangle = \langle u,v \rangle$$ for all $g\in\mathbb{Z}_3, u,v\in\mathbb{C}^2$.
(b) Describe all $\rho$-invariant inner products on the vector space $\mathbb{C}^2$ explicitly, in terms of the coordinates of vectors.
I've done (a) but how do I do (b)?