I have learned that the Hopf fibration is a principal $S^1$-bundle over $S^2$. The $S^1$-action on $S^3\subseteq\mathbb{C}^2$ is given by $$w\cdot(z_1,z_2)=(wz_1,wz_2).$$ Now let us consider the representation $$\rho:S^1\to \text{Diff}(S^1), \quad w\mapsto \text{left multiplication by }w^n.$$ What is the associated bundle $E=S^3\times_{S^1}S^1$?
My thought: For $[z_1,z_2,w]\in E$, we have $$[z_1,z_2,w]=[z_1w^{1/n},z_2w^{1/n},1].$$ Does this mean that $E\cong S^{\color{red}3}$ as fibre bundle? If not, where am I wrong?