Let $f$ be a cusp form and $\pi_f$ be the corresponding cuspidal representation of ${\mathrm{GL}}_2(\mathbb{A}_\mathbb{Q})$. If $\chi$ is a Dirichlet character modulo $M$, then we know that $f \otimes \chi$ corresponds to the cuspidal representation $\pi_f \otimes (\chi\cdot \det)$.
Now if we start with a cuspidal representation of ${\mathrm{GL}}_2(\mathbb{A}_\mathbb{Q})$, say $\pi$, then from Theorem 5.19 of this book by Gelbart, we know that $\pi$ corresponds to some newform $f$. Now if we twist $\pi$ by some Hecke character $\psi$ which is not a Dirichlet character, i.e. if we consider $\pi \otimes \psi$ , then what will be the corresponding cusp form?
In other words, I want to know if we can twist a newform by a non-Dirichlet Hecke character.
If not, then what is happening here? I am certainly missing some major point in the Theorem 5.19.
Any help regarding this would be really appreciated.
Thanks in advance.
There is really nothing happening here. This becomes a great deal more transparent if you take the view that a modular newform (holomorphic or Maass) corresponds to an equivalence class of automorphic representations, differing by twists by the norm character.
For any $t$, you can cook up from $f$ a representation of $GL_2(\mathbb{A})$ whose central character restricts to $x \mapsto x^t$ on $\mathbb{R}_{>0}^\times$, and all of these (or all the unitary ones, if you prefer, which correspond to taking $t$ totally imaginary) have an equal claim to being "the" automorphic representation associated to $f$. Choosing different $t$-values just corresponds to twisting $\pi$ by some power of $\|\cdot\|$, giving you a different representation in the same equivalence class. (Every now and again, arguments erupt on math.SE or MathOverflow as to which is the "correct" value of $t$; and these arguments are about as conclusive as those of the Little-Endians and Big-Endians in Gulliver's Travels.)
Now, if $\psi$ is a Grössencharacter $\mathbb{Q}^\times \to \mathbb{A}_{\mathbb{Q}}^\times$, then $\psi |_{\mathbb{R}_{> 0}^\times}$ has to be of the form to $x \mapsto x^s$, for some complex $s$. In other words $\psi$ has the form $\|\cdot\|^s \psi_0$, for some $\psi_0$ whose restriction to $\mathbb{R}_{> 0}^\times$ is trivial.
The Grössencharacters trivial on $\mathbb{R}_{> 0}^\times$ correspond exactly to Dirichlet characters, and it sounds to me as if you are already happy with those. That just leaves powers of the norm character, and the only effect of these is to change the $t$ parameter, giving you a different representation in the same equivalence class.
(It is only once you consider nontrivial number fields – and, if you believe Leopoldt's conjecture, non-totally-real number fields – that you start to see "interesting" Grössencharacters, i.e. ones which aren't the product of a finite-order thing and a power of the norm map.)