I am thinking of Eisenstein series, $$G_k(\tau) = \sum_{(c,d)\in {\mathbb{Z}}^2-\{(0,0)\}}\frac{1}{(c\tau+d)^k}, \tau \in \mathbb{H} $$ because we don't sum over $(0,0)$, so I'd like to call $(0,0)$ a pole. Am I right or wrong?
My other intuition is the $j$-function because it's q-expansion shows it has pole at $\infty$; BUT some texts don't consider $j$-function as a modular form, they only consider it a modular function. So, if these two are not the right examples, could someone please direct me to the right ones?
Thank you very much!