Let $\Gamma$ be a normal subgroup of finite index of the modular group $PSL(2,\mathbb{Z})$.
Let $\mathbb{H}$ be the upper half-plane.
Let $f$ be a holomorphic complex function defined on the upper half-plane $\mathbb{H}$, i.e., $f:\mathbb{H} \rightarrow \mathbb{C}$.
Then, $f(z)dz$ can be written as follows $$f(z)dz=(u(x,y)+iv(x,y))(dx+idy)$$ Therefore, the real part of $f(z)dz$, denoted by $h$, is $u(x,y)dx-v(x,y)dy$, i.e., $$h=u(x,y)dx-v(x,y)dy$$
It is well-known that $h$ is harmonic $1-$ form on $\mathbb{H}$, and it induces a harmonic $1-$ form on the Riemann surface $\mathbb{H} / \Gamma$, denoted by $w$.
So $w$ assigns to each point $x_0 \in \mathbb{H} / \Gamma$ a linear map $w_{x_0}: T_{x_0} (\mathbb{H} / \Gamma) \rightarrow \mathbb{R}$.
Where $T_{x_0} (\mathbb{H} / \Gamma)$ is the tangent space of $\mathbb{H} / \Gamma$ at the point $x_0$.
Am I correct?
Can the space $T_{x_0} (\mathbb{H} / \Gamma)$ be described better? (Is it $\mathbb{C}$?).