I am beginning to learn about automorphic forms, and stay perplex concerning the two languages of "forms" versus "representations" often used at the same time. As far as I understand,

- a modular/Maass form generates an automorphic representation (by considering the space of its right translations)
- conversely, an automorphic representation gives a unique automorphic form (as a newvector, i.e. a nontrivial element in $\pi^K$ which is one dimensional for a suitable (?) compact subgroup $K$)

I would like to understand, where does the distinction between modular form and Maass form appear in the representation language?

Thanks in advance for any reference or explanation