Question

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.

Some examples

(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.

Answer 1

Characters of rational vertex operator algebras tend to yield modular functions. This is due to the space of torus partition functions in a chiral conformal field theory being a complex moduli invariant. The standard example is the monster vertex algebra, whose character is j-744. Other examples come from lattice CFTs (presumably describing a bosonic string propagating in a torus), and have the form of a theta function divided by a power of eta. The characters are never Hecke eigenforms, because of the pole at infinity, but traces arising from higher-weight vectors may be. In some cases, the vertex algebra structure is supposed to arise from geometry of a target space, so this phenomenon may be related to Hirzebruch-Zagier (number 5).

Characters of highest weight representations of affine Kac-Moody algebras yield modular forms. One can reasonably argue that this is a special case of the previous paragraph, since (I think) they come from Wess-Zumino-Witten. The Weyl-Kac character formula for such representations is one way to get Macdonald identities, and the smallest case (trivial rep of affine sl₂) yields the Jacobi triple product.

Answer 2

Another example: various subsequences of the integer partition function p(n) occur as coefficients of (sometimes half-integer weight) modular forms. One reference is Ahlgren and Boylan "Arithmetic properties of the partition function".