It occurs more and more that I ask a question on math stackexchange and then realize that it is more appropriate to mathoverflow. Hopefully this reflects well on myself... In any case, I copy here word for word my question from stackexchange, which received no answers, and I will erase it from there.

In a previous question of mine from a long time ago Matt E gave an excellent exposition to Langlands which has benefited me immensely. Now that I have matured a little bit, I wish to know certain things in more detail.

In the answer to my previous question, Matt E detailed how newforms are related to motives. Newforms are classifies as either classical Atkin Lehner newforms of some weight $k$; and Maass forms of weight $\lambda=\frac14$. Classical Atkin Lehner newforms of weight $k$ should correspond to motives with degree of purity $k-1$, and Maass forms of weight $\lambda=\frac14$ should correspond to degree of purity $0$ motives. (So degree of purity $0$ motives are those that come from classical newforms of weight $1$ and Maass forms.)

A classical Atkin Lehner newform is a cuspform in some $\Gamma_0(N)$ for some $N$, which is a simultaneous eigenvalue of all Hecke operators, such that $N$ is minimal with respect to the associated system of eigenvalues.

My question is: what, if any, is the role played by other congruence and non-congruence subgroups other than $\Gamma_0(N)$? Do they fit into the Langlands philosophy in some way?

by definitiondefined using congruence conditions. There is a story involving non-congruence subgroups, but it is somehow another layer of complexity on top of the "usual" Langlands programme. $\endgroup$