Cusp form growth condition

Question

Zagier gives an equivalent definition of a cusp form, being a modular form with Fourier expansion $f(z)=\sum_{n=0}^{\infty}a(n)e^{2\pi i n z}$ with $a(0)=0$. This is apparently equivalent to $|f(z)|<M\Im(z)^{-k/2}$ for some $M\in\mathbb{R}$. I see that if $|f(z)|<M\Im(z)^{-k/2}$ then we must have $a(0)=0$ since $\sum_{n=1}^{\infty}a(n)e^{2\pi i n z}$ vanishes as $\Im(z)\rightarrow\infty$. But I don't see why the reverse implication holds. We'd need to bound the $a(n)$?