Let $\mathbb{H}$ be the upper half plane, and $\text{SL}_2(\mathbb{Z})$ acts on $\mathbb{H}$ by linear transformation $$\begin{pmatrix} a & b \\ c & d \end{pmatrix}\cdot z = \frac{az+b}{cz+d}.$$
I know that one can give a complex structure to $\text{SL}_2(\mathbb{Z})\backslash \mathbb{H}$, making this a Riemann surface, so that the quotient map $\mathbb{H} \to \text{SL}_2(\mathbb{Z})\backslash \mathbb{H}$ is holomorphic. However, this map is not a covering map (look at $i\in \mathbb{H}$ locally, then this map looks like $z\mapsto z^2$), so $\mathbb{H}$ cannot be seen as a universal cover of $\mathbb{H}$ via this map.
So my question is: What is the universal cover of $\text{SL}_2(\mathbb{Z})\backslash \mathbb{H}$? By uniformization theorem, is this universal cover holomorphic (after giving it a complex structure via the covering map) to $\mathbb{C}$, $\mathbb{H}$ or $\mathbb{CP}^1$?