I have one issue with the Jacquet Langlands correspondance. The Weyl law for $H$ modulo a congruence subgroup and the Weyl law for cocompact groups are different. So why does this not contradict this functoriality? What am I missing?
I have not yet studied the Jacquet Langlands correspondance explicitely yet. How explicit are the lifts, about the level etc.? I know that there is not an expansion formula for cocompact groups available as we have it for groups with an parabolic element.
Update: After a reading a little bit, I found a paper which focuses exactly on the first part of the question and also gives references for the second part of the question, i.e.
Risager, Morten S. Asymptotic densities of Maass newforms. J. Number Theory 109 (2004), no. 1, 96–119.