Let $\mathbb{H}=\{z \in \mathbb{C}: \operatorname{Im}(z)>0\}$. A biholomorphic map $F: \mathbb{H} \to \mathbb{H}$ is represented by
\begin{align*}
F(z)=\frac{az+b}{cz+d}
\end{align*}
for some $a,b,c,d \in \mathbb{R}$ with $ad-bc>0$. Two such coefficient lists define the same automorphism exactly when they differ by multiplication by a nonzero real scalar.