The modular and entire modular forms for a subgroup of finite index of the modular group $PSL(2,\mathbb{Z})$

Question

Let $PSL(2,\mathbb{Z})$ be the modular group and $\Gamma$ be a subgroup of finite index of the modular group $PSL(2,\mathbb{Z})$.

I think that it is well-known that a function $f:\mathbb{H}\rightarrow \mathbb{C}$ is called a modular form for the subgroup $\Gamma$ of weight $k$ if:

1. $f(g(z)) = (cz+d)^k f(z)$, for all $z \in \mathbb{H}, g\in \Gamma$.
2. $f$ is holomorphic in $\mathbb{H}$.
3. $f$ is holomorphic at the cusps of $\Gamma$.

Now, How is the entire modular form for the subgroup $\Gamma$ defined?