Is there a choice of the basis of $\mathbb{C}Z_3$ such that the regular representation of $Z_3$ is $$\begin{pmatrix} 1&0&0\\ 0&1&0\\0&0&1 \end{pmatrix},\begin{pmatrix} w&0&0\\ 0&1&0\\0&0&1 \end{pmatrix} \text{and}\begin{pmatrix} w^2&0&0\\ 0&1&0\\0&0&1 \end{pmatrix}$$ where $w$ is a primitive $3$rd root of unity.
I haven't found one for this. Based on cyclic I do have found a basis for: $$\begin{pmatrix} 1&0&0\\ 0&1&0\\0&0&1 \end{pmatrix},\begin{pmatrix} 1&0&0\\ 0&w^2&0\\0&0&w \end{pmatrix} \text{and}\begin{pmatrix} 1&0&0\\ 0&w&0\\0&0&w^2 \end{pmatrix}.$$