Given an algebraic number field $F$, which we may as well take to be Galois over $\mathbb{Q}$, we denote by $S_F$ the set of rational primes that split in $F$. Sets of the form $S_F$ are called Galoisian. At some point, there was a discussion

http://mathoverflow.net/questions/11688/why-do-congruence-conditions-not-suffice-to-determine-which-primes-split-in-non-a

of the fact that the *abelian Galoisian sets*, that is, $S_F$ corresponding to $F$ abelian over $\mathbb{Q}$, are exactly the sets of primes defined by congruence conditions.

A while later, Matthew Emerton gave this nice answer

http://mathoverflow.net/questions/11747/galoisian-sets-of-prime-numbers

to a question of Chandan Singh Dalawat about non-abelian Galoisian sets. 

I made a comment there I thought I would upgrade to a question. As Matthew points out,  Neukirch's thought that the Langlands program would provide a *characterization* of all Galoisian sets is probably meant as a metaphor for some other process. However, I couldn't help but hope that the characterization could be taken literally, at least for some special families. For example, we will refer to a number field $F$ as being of $GL_2$ type if it is the fixed field of
$Ker(\rho)$, where
$$\rho: Gal(\bar{\mathbb{Q}}/\mathbb{Q})\rightarrow GL_2(\mathbb{C})$$
is an irreducible two-dimensional Artin representation of $Gal(\bar{\mathbb{Q}}/\mathbb{Q})$*. 
 Now call a set of primes *a solvable $GL_2$ Galoisian set* if it is of the form $S_F$ for some solvable extension $F$ of $GL_2$-type.

The question then  is: can one use the Langlands program (or anything else) to give a sensible characterization of solvable $GL_2$ Galoisian sets?

There is an awkward sort of guess I could make, but I would rather see what the experts say.

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*The idea that we should simply organize fields in this manner correspending to representations is perhaps a valuable perspective coming out of the Langlands program.