Note: I've revised the question just a little bit in the hope of making it easier.
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
Why do congruence conditions not suffice to determine which primes split in non-abelian extensions?
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
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 remark that the Langlands program provides 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 IOne could obviously change this question in any way that would make it more tractable. One could try to characterize, but I would rather see whatfor example:
-Solvable $GL_2$ Galoisian sets, where the experts say$GL_2$-field $F$ is further required to be solvable;
-Odd $GL_2$ Galoisian sets: $S_F$ where $F$ is the fixed field of a representation $\rho_f$ arising from a holomorphic modular form $f$ of weight one;
-Odd $GL_2$ Galoisan sets of conductor $N$, where we further require the form $f$ to have level $N$;
and so on. The last case probably admits a tautological answer of sorts, in that we can in principle list the finitely many forms (sorted by Dirichlet characters $\epsilon$), and then make some statement about the $p$'s where
$$X^2-a_pX+\epsilon(p)=(X-1)^2.$$
Is it entirely unreasonable to hope for something more compact?
*The idea that we should simply organize fields in this manner corresponding to representations is perhaps a valuable perspective coming out of the Langlands program.