Orbifold of the three-sphere

Question

Think of the three-sphere as given by $\lbrace|z|^2+|w|^2=1, \;z,w\in \mathbb{C}^2\rbrace$. We can regard it in terms of Hopf coordinates \begin{align*} z&= \cos(\theta/2)e^{i(\phi+\psi)}\\ w&= \sin(\theta/2)e^{i\psi} \end{align*} where $0\leq\theta\leq \pi$ and $0\leq \phi,\psi<2\pi$. Now I want to consider the orbifold obtained when we divide $S^3$ by the following action: $$(z,w)\mapsto (z,e^{2\pi i/k}w)$$ for $k$ some integer. In terms of the Hopf coordinates, this identifies $$(\phi,\psi)\sim (\phi-2\pi /k,\psi+2\pi /k)$$ Is it correct that the resulting space is a Seifert fibration with $$(b,g;(p_1,q_1),(p_2,q_2))=(1,0;(k,-1),(-k,1))$$ where $b$ is the Euler number of the circle fibration, $g$ the genus of the base and $(p_i,q_i)$ label the fixed points of the action described above. Also, applying the same orbifolding to the Lens spaces $L(n,1)$, is it correct that the resulting orbifolds are the Seifert fibrations $$(b,g;(p_1,q_1),(p_2,q_2))=(n,0;(k,-n),(-k,n))$$