This is a question in Serre's book "A Course in Arithmetic". He said "We can thus identify modular functions of weight $k$ with some lattice functions of weight $k$."
I am explain my question as follows.
He definites "modular functions" to require that it must meromorphic
at upper half plane $H$ and $\infty$,
of course,
satisfing
$f\left(\frac{az+b}{cz+d}\right)=(cz+d)^{k}f(z).$
However,
in his definition of lattice function,
he hasn't mentioned "meromorphic" condition.
Thus, $f$ may not a meromorphic function. What I mean is that we cannot regard a lattice functions of weight $k$ as a modular functions of weight $k$. Am I wrong...?