Do all cuspidal automorphic representations of $\operatorname{GL}_2(\mathbb A_{\mathbb Q})$ come from Maass or holomorphic cusp forms?

Question

A normalized cuspidal newform $f$ (either holomorphic or Maass) can be identified with a function on $\phi: \operatorname{GL}_2(\mathbb Q) \backslash \operatorname{GL}_2(\mathbb A_{\mathbb Q})$, and it generates an irreducible cuspidal automorphic representation in some $L^2$-space. The central character of the corresponding representation depends on the some choice made in the transfer from $f$ to $\phi$.

Do these representations exhaust the cuspidal spectrum of $\operatorname{GL}_2(\mathbb A_{\mathbb Q})$?