Fields of rationality as a notion of automorphic size

Question

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).

(A) Do the two notions agree in the case of Maass forms?

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:

(B) Is there any result in the weight aspect?

(C) Is $d(\pi)$ always finite?

(D) Is there any infinite family with constant degree of field of rationality? What can be said about such families?

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.

Answer 1

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]

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.