This seems to say that weight 2 weakly holomorphic modular forms have zero constant term. I don't quite follow the argument though... But I am wondering if this is also true for $M_{2}^{!}(N)$. That is, must $f \in M_{2}^{!}(N)$ have zero constant term in its Fourier expansion?
Edit: I am following an argument about a weight $2$, level $32$ weak form $f$. Let's say $f \in M^{!}_{2}(32)$ vanishes at all cusps except $\infty$. The meromorphic differential $f(z)dz$ is holomorphic everywhere except at $\infty$. The sum of residues of a meromorphic differential is $0$. Moreover, the residue at $\infty$ of any meromorphic differential $h(z)dz$, where $h$ is any weight $2$ form, is a multiple of its constant term in its $q$-expansion (why?). Thus the residue at $\infty$ must be $0$ (why?).