I want to interpret the degree of the field of rationality of an automorphic form as a notion of size, analogously to the conductor, and this question is about the possible obstructions to do so. The results I know are limited to the case of Hecke cusp forms, I hense recall them and raise the natural questions and results to Maass forms and automorphic representations.


**1. The case of Hecke cusp forms.** Let $f$ be a cuspidal Hecke eigenform. Its field of rationality $\mathbf{Q}(f)$ is the extension of $\mathbf{Q}$ generated by its Fourier coefficients, that is to say
$$\mathbf{Q}(f) = \mathbf{Q}(a\_1(f), a\_2(f), a\_3(f), \ldots). \qquad (1)$$


It is known that the degree of the field of rationality grows with the level (in the sense that the proportion of $f$ with field of rationality of bounded degree converges to zero when the level grows).


**2. The case of automorphic representations.** I am wondering about what can be said of an analogous notion for automorphic representations. Let $\pi\_v$ be an admissible representation of $GL(2, F\_v)$ for a local field $F\_v$, its field of rationality is defined as
$$\mathbf{Q}(\pi\_v) = \{ \sigma \in Aut(\mathbf{C}) \ : \ {}^\sigma \pi \simeq \pi \}.$$


For an automorphic representation $\pi$ of $GL(2, \mathbf{A})$, decomposed by Flath theorem as $\pi = \otimes\_v \pi\_v$, the field of rationality of $\pi$ is
$$\mathbf{Q}(\pi) = \prod\_v \mathbf{Q}(\pi\_v). \qquad (2)$$


**3. The case of Maass forms.** These two notion agree in the case of Hecke cusp forms. I wonder what can be said in the case of Maass forms: a field of rationality can be defined by its coefficients as in (1), and also by the attached representation as in (2).



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> (A) Do the two notions agree in the case of Maass forms?
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**4. A kind of automorphic size.** As already stated, in the case of cusp forms the degree of the field of rationality essentially grows with the level (Serre, Shin-Templier, Binder for references). This endows
$$d(\pi) = [\mathbf{Q}(\pi) : \mathbf{Q}],$$


with a size flavor. However, here are some natural questions in this direction:



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> (B) Is there any result in the weight aspect?
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> (C) Is $d(\pi)$ always finite?
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> (D) Is there any infinite family with constant degree of field of rationality? What can be said about such families?
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I apologize for these maybe loosely related questions, it appeared to me that they are relevant in this spirit of "size", and I wished to asked them together.



Fields of rationality as a notion of automorphic size
I haven't thought about the case of Maass forms, but I can tell you what is expected about the weight.


*Maeda's conjecture* implies that all cuspidal eigenforms in $S\_k(1)$ are conjugate, so $\bar d(\pi)$ should be $\dim S\_k(1)$, where $\bar d$ denotes the degree of the Galois closure. In particular, the answer to (D) should be no for level 1, but I don't think anyone knows how to prove such a statement. See also this question: [modular eigenforms with integral coefficients [Maeda's Conjecture]](https://mathoverflow.net/q/11677/6518)


A generalization of Maeda's conjecture was proposed by Tsaknias for more general level. For instance, say $N$ is squarefree. Then if you look at newforms of level $N$, you can distinguish Galois orbits by looking at Atkin-Lehner sign patterns (of which there are $2^m$, $m =$ number of primes dividing $N$). The conjecture is that for $k$ large all forms with the same Atkin-Lehner sign pattern are conjugate.


