I have recently started studying the automorphic science and find it somewhat hard to form intuition. Can we have a list of examples of automorphic representations that you usually use to test a new idea, or that are counterexamples to a statement that might look reasonable to a beginner but is actually false? Inspired by this question.

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    $\begingroup$ GL(1), GL(2) and endoscopic lifts, like Saito-Kurokawa. The latter examples disproved the original generalized Ramanujan conjecture. $\endgroup$ – Kimball May 15 at 13:31

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