What are the simplest numerical examples of even dihedral (resp. tetrahedral, resp. octahedral, resp. icosahedral) representations
$\rho:\mathrm{Gal}(\bar{\mathbf{Q}}|\mathbf{Q})\to\mathrm{GL}_2(\mathbf{C})$
and their associated Maaß forms $f_\rho\ $ ? The word simplest can be taken to mean that the conductor of $\rho$ is small, or a small prime.
[Serre 1977] and [Buzzard 2012] provide many simple examples of odd Artin representations $\rho$ of degree $2$ and the associated cuspidal modular forms of weight $1$. For example, the splitting field of $T^3-T-1\ $ gives rise to an odd dihedral representation of conductor $23$ whose associated weight-$1$ modular form is
$$
q\prod_{n>0}(1-q^n)(1-q^{23n}),
$$
as discussed by Emerton in MO11747. For the simplest examples when the image of $\rho$ in $\mathrm{PGL}_2(\mathbf{C})$ is isomorphic to $\mathfrak{A}_4$, $\mathfrak{S}_4$ or $\mathfrak{A}_5$, see [Buzzard 2012].
I'm looking for something similar for even representations. I'm aware of [Vignéras 1985], so my real question is whether there has been further progress in constructing even Artin representations $\rho$ of degree $2$, or the Maaß forms which are supposed to correspond bijectively to such $\rho$ ?
[Buzzard 2012] Computing weight one modular forms over $\mathbf{C}$ and $\overline{\mathbf{F}}_p$. arXiv:1205.5077.
[Serre 1977] Modular forms of weight one and Galois representations. Algebraic number fields: L-functions and Galois properties (Proc. Sympos., Univ. Durham, Durham, 1975), pp. 193–268. Academic Press, London.
[Vignéras 1985] Représentations galoisiennes paires. Glasgow Math. J. 27, 223–237.
Addendum. (2012/06/14) Came across the short write-up Explicit Maass Forms by Kevin Buzzard which contains some nice examples.
Best Answer
I'll leave the dihedral and octahedral case to others, but for the tetrahedral ($A_4$) and icosahedral ($A_5$) case, I can give some answer.
For the tetrahedral case, the smallest conductor is 163. See my question: Does anyone want a pretty Maass form?
I have some (not very well documented) code to compute Artin L-function coefficients for this even tetrahedral Galois representation and thus the (provably, by Langlands) Maass form. I've posted this code on my webpage http://people.ucsc.edu/~weissman/
For the icosahedral case, a totally real $A_5$ extension of the rationals is given by the splitting field of $x^5 + 5 x^4 - 7 x^3 - 11 x^2 + 10 x + 3$. In my 1999 undergraduate senior thesis, at http://people.ucsc.edu/~weissman/MWSenThesis.pdf, I found some mild numerical evidence that the associated degree-3 L-function is entire. I can't find it written there, but I'm guessing it's the first, or among the first, $A_5$ extensions of $Q$. I would have chosen something of minimal conductor, to minimize the compute-time. I think I found this by looking in tables from J. Buhler's thesis.
Note that I didn't lift the projective representation $Gal \rightarrow PGL_2(C)$ to an honest 2-dimensional representation. This is a bit subtle, and I wasn't capable of that work at the time. Using the 3-dimensional representation (since $A_5$ has a faithful 3-dim representation) avoids this issue, capturing the adjoint square lift of the putative Maass form. I'd guess that lifting the projective representation would be possible now, if someone wanted to do the work.