Let $f$ be a periodic holomorphic function on the upper half plane $\mathbb H$. Here periodic means $f(z+1)=f(z)$ for all $z \in \mathbb H$.
Then $f$ is equal to its Fourier series $$ f(z)=\sum_{n \in \mathbb Z} c_n q^n, \quad q=e^{2\pi i z}. $$
Suppose the Fourier series of $f$ has the form
$$ \sum_{n \geq n_0} c_n q^n $$
for some $n_0 \in \mathbb Z$.
Is it true that
$$ \sum_{n \geq 1} c_n q^n \ll e^{-\epsilon y} $$ for some $y$, where $z=x+iy$?
This question is originally from the following statement : 'A weakly holomorphic modular form is a harmonic maass form.' For the original question, see https://mathoverflow.net/questions/357309/a-weakly-holomorphic-modular-form-is-a-harmonic-maass-form