I'm reading a quick example (Example 12.13 of Topological Manifolds by John Lee) of the construction of the lens space $L(n,m)$. Basically, let $$S^3=\{(z_1,z_2)\in\mathbb{C}^2:|z_1|^2+|z_2|^2=1\}$$ and fix $1\leq m<n$ coprime. Define a $\mathbb{Z}/(n)$ action on $S^3$ by $$ k\cdot (z_1,z_2)=(e^{2\pi ik/n}z_1,e^{2\pi i km/n}z_2) $$ which is a free and proper action. Then the orbit space $S^3/(\mathbb{Z}/(n))$ is a compact $3$-manifold with fundamental group $\mathbb{Z}/(n)$. This orbit space is the lens space $L(n,m)$.
However, I don't see anything special about $m$ other than it's an integer coprime to $n$, which is used to show the group action is free. Then I think $L(n,m)=L(n,r)$ for any integers $m,r$ coprime to $n$, which makes me question why the $m$ is included in the notation.
Is there something special about the particular $m$ I'm missing?