Global irreducible admissible representations analogue

Question

Let ${A}_{\mathbb{Q}}$ denote the adele over the rational numbers. Then it is known that cuspidal modular forms of level $N$ correspond to some unitary automorphic representation of $\operatorname{GL}_2({A}_{\mathbb{Q}})$. My question is that are there some functions $f$ on upper half plane of the form $$f(z)=\sum\limits_{n=1}^{\infty}a_ne^{2\pi inz}$$ satisfying suitable conditions which correspond to an (irreducible) admissible (not necessarily automorphic) representations of $\operatorname{GL}_2({A}_{\mathbb{Q}})$. I have not been able to find anything about this in the literature. Any help is deeply appreciated.