# Identifying Modular Functions of Weight $2k$ with Lattice Functions of Weight $2k$

I have been reading Serre's 'A Course in Arithmetic'. In section $$2.2$$ he shows that lattice functions $$F$$ of weight $$2k$$ are in one-to-one correspondence with modular functions of the same weight, by defining for each lattice function $$F$$, the function $$f: \mathbb{H} \longrightarrow \mathbb{C}$$ given by $$f(\frac{\omega_1}{\omega_2}) := \omega_2^{2k}F(\omega_1, \omega_2) = F(\Gamma(\omega_1, \omega_2))$$. While it is clear (by the defining property $$F(\lambda \Gamma) = \lambda^{-2k} F(\Gamma) \cdots \hspace{1.0mm}(1)$$ of the lattice function $$F$$) that $$f$$ is well-defined and the modularity property of $$f$$ also follows by the invariance of $$F$$ under $$SL_2(\mathbb{Z})$$-action (a consequence of the well-definedness of $$F$$ as a lattice function), it is not clear to me why $$f$$ must be meromorphic on $$\mathbb{H} \cup \{\infty\}$$ (for that is how a modular function has been defined by him), since all we know about $$F$$ (which is the only thing used to defined $$f$$) is the property $$(1)$$. Is this a consequence of some property of lattice functions? Thanks in advance.