The *reciprocity* of the title is the following not completely well-posed problem:

Fix $P(X)$ a monic irreducible polynomial of degree $n$, with coefficients in $\mathbb Z$. "Describe" (in some sense) the set of prime $p$ such that $P \mod p$ has a given reduction (for example is irreducible, or on the opposite is a product of $n$ linear factors, etc.).

As is well-known, when one root (hence all roots) of $P$ happens to be a cyclotomic integer (in particular in the very special case where $P$ is quadratic), then the problem has a precise solution, called Artin's reciprocity law, which is the core of class field theory. The sets of primes $p$ such that $P$ has a given reduction are then given by congruences modulo fixed integers satisfied by $p$, that is those sets are union of "set primes in artithemtic sequence". The Galois group of $P$ is abelian in this case.

As the opposite of the complexity spectrum, there are polynomial $P(X)$ of degree $n \geq 5$, whose Galois group is big, say $S_n$, or $A_n$ in particular not solvable. In this case, it is my understanding that our only hope to find a reciprocity law in general is by making huge progress in the Langlands program.

But is it possible, in some special case (for polynomials $P$ of Galois group $S_n$ say satisfying somme assumtions, or for an explicit family of polynomials of variable degree $n \geq 5$, or even for just one explicit polynomials) where some kind of reciprocity law has been worked out (even only partially) by some method that does not involve the Langlands program (i.e. modular forms, automorphic forms, etc.) ?

issoluble, but I am pretty sure that Euler didn't know about the Langlands program. On the other hand of course this doesn't answer the question, because the weight 1 forms that you can access using theta series are precisely those ones giving dihedral Galois reps. $\endgroup$8more comments