Given $N$, can we always find a non-constant weakly holomorphic modular form of level $N$ which is holomorphic at all cusps except possibly at $\infty$?
For example, $S_{4}(16)$ is $1$-dimensional so its generator is that non-constant (weakly) holomorphic modular form I'm looking for. On the other hand, $S_{3}(16)$ has dimension $0$. So what can I say about the space $S_{3}^{\#}(16) := \{f \in M^{!}_{3}(16) : f\text{ holomorphic at all cusps except possibly at } \infty\}$? That is, how do I know if it's non-trivial? Or how can I search for a non-constant element in such a space?