## Number theoretic sequences and Hecke eigenvalues

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.

-
 Theta functions aren't restricted to sums of squares. You can enumerate representations of natural numbers as the output of an arbitrary positive definite quadratic form. – S. Carnahan♦ Oct 28 2009 at 6:55

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.

-

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".

-