What are some number theoretic sequences that you know of that occur as (or are closely related to) the sequence of Fourier coefficients of some sort of automorphic function/form or the sequence of Hecke eigenvalues attached to a Hecke eigenform?
I know many such sequences, but am always looking for more.
(1) The sequence a(n) deriving from the traces a(p) of the Frobenius elements in a Galois representation (Langlands reciprocity conjecture)
(2) Number of representations of a natural number as a sum of k squares (theta functions)
(3) The sum of powers of divisor functions (Eisenstein series)
(4) The central critical values of L-functions attached to all quadratic twists of a Hecke eigenform (Kohnen, Waldspurger)
(5) Intersection numbers of certain subvarieties of Hilbert modular surfaces (Hirzebruch-Zagier)
I'll end with a question that is ill-posed but nevertheless very interesting (at least to me personally): why do so many familiar and yet diverse sequences appear in this fashion? Note that many of them have a history of study that precedes the recognition that they are essentially coefficients of automorphic functions/forms.