Basically, as I know, we know almost nothing about Maass forms. For example, Cohen constructed first (maybe not) example of a Maass cusp form by using one of Ramanujan's $q$-series, as a non-definite theta series. (This is from his Inventiones paper, *$q$-identities for Maass waveforms*.) Also, this article from Buzzard gives some explicit examples, both "algebraic" (corresponds to even 2-dimensional Galois representations) and "non-algebraic" (something that mysterious?)
Algebraic ones has an eigenvalue $1/4$, while non-algebraic ones have (conjecturally) transcendental eigenvalues (such as
$$
\frac{1}{4} + \frac{\pi^{2}}{4\log^{2}(\sqrt{2} - 1)}
$$
Also, this paper provides an algorithm to compute eigenvalues effectively, and the first eigenvalue for $\mathrm{SL}_{2}(\mathbb{Z})$ is $\lambda = \frac{1}{4} + r^{2}$, where $r = 9.5337...$.

I think the eigenvalues should be some special numbers, which can be transcendental, but maybe a little *algebraic*. More precisely, I hope that these numbers are related to *periods*. Periods are defined in Zagier-Kontsevich's article (this) as numbers that can be obtained by integral of rational functions over a domain defined by inequalities of rational coefficient polynomials. For example, any algebraic numbers, $\pi, \log 2, \zeta(3)$ are periods. Especially,
$$
\log(\alpha) = \int_{1}^{\alpha} \frac{1}{x} dx
$$
is a period for any $\alpha\in \overline{\mathbb{Q}}$. Periods form a proper subring of $\mathbb{C}$ that contains $\overline{\mathbb{Q}}$, denoted by $\mathcal{P}$, and $\mathcal{K} = \mathrm{Frac}(\mathcal{P})$ is a field between $\overline{\mathbb{Q}}$ and $\mathbb{C}$. So here is my conjecture:

If $\lambda$ is an eigenvalue of a Maass waveform (cusp form) on $\Gamma_{0}(N)$, then $\lambda \in \mathcal{K}$.

Obviously, there's no reason that this conjecture is true. The reason I belive this is because the eigenvalues shouldn't be just random transcendental numbers. To *prove* this conjecture, the first thing we have to figure out is the exact value of the smallest eigenvalue for $\mathrm{SL}_{2}(\mathbb{Z})$, which is approximately 91.14. Thanks in advance.

p.s. According to this question, if we assume some conjectures about motivic stuff, then $\mathcal{P}$ is not a field.

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