In short: what does Labesse-Langlands say?

Slightly more precise: what are the cuspidal automorphic representations of $SL_2(\mathbf{A}_{\mathbf{Q}})$, together with multiplicities? Let's say that I have a complete list of the cuspidal automorphic representations of $GL_2/\mathbf{Q}$ and I want to try and deduce what is happening for $SL_2$. I am looking for "concrete examples of the phenomena that occur".

Now let me show my ignorance more fully. My understanding from trying to read Labesse-Langlands is the following. The local story looks something like this: if $\pi$ is a smooth irreducible admissible representation of $GL(2,\mathbf{Q}{}_p)$ then its restriction to $SL(2,\mathbf{Q}{}_p)$ is either irreducible, or splits as a direct sum of 2 non-isomorphic representations, or, occasionally, as a direct sum of 4. One interesting case is the unramified principal series with Satake parameter $X^2+c$ for any $c$; this splits into two pieces (if I've understood correctly) (and furthermore these are the only unramified principal series which do not remain irreducible under restriction). Does precisely one of these pieces have an $SL(2,\mathbf{Z}{}_p)$-fixed vector? And does the other one have a fixed vector for the other hyperspecial max compact (more precisely, for a hyperspecial in the other conj class)? Have I got this right?

Packets: Local $L$-packets are precisely the J-H factors for $SL(2,\mathbf{Q}{}_p)$ showing up in an irreducible $GL(2,\mathbf{Q}{}_p)$-representation. So they have size 1, 2, or 4.

Now globally. a global $L$-packet is a restricted product of local $L$-packets (all but finitely many of the components had better have an invariant vector under our fixed hyperspecial max compact coming from a global integral model). Note that global automorphic $L$-packets might be infinite (because $a_p$ can be 0 for infinitely many $p$ in the modular form case). My understanding is that it is generally the case that one element of a global $L$-packet is automorphic if and only if all of them are, and in this case, again, generally, each one shows up in the automorphic forms with the same multiplicity. What is this multiplicity? Does it depend?

Finally, my understanding is that the above principle (multiplicities all being equal) fails precisely when $\pi$ is induced from a grossencharacter on a quadratic extension of $\mathbf{Q}$. In this case there seems to be error terms in [LL]. Can someone explain an explicit example where they can say precisely which elements of the packet are automorphic, and what the multiplicities which which the automorphic representations occur in the space of cusp forms?

I find it very tough reading papers of Langlands. My instinct usually would be to press on and try and work out some examples myself (which is no doubt what I'll do anyway), but I thought I'd ask here first to see what happens (I know from experience that there's a non-zero chance that someone will point me to a website containing 10 lectures on Labesse-Langlands...)

Edit: I guess that there's no reason why I shouldn't replace "cuspidal" by "lies in the discrete series" with the above (in the sense that the questions then still seem to make sense, I still understand everything (in some sense) for $GL_2$ and I still don't know the answers for $SL_2$)