  We study graded traces of vectors in free bosonic vertex operator algebras
and lattice vertex operator algebras. We show in particular that trace
functions in these two theories always have the shape f(q)/\eta(q)^d where f(q)
is quasi-modular in the case of d free bosons, and modular (i.e., a sum of
holomorphic modular forms of various weights) in the case of theories based on
a lattice L of rank d. We also show how spherical harmonic polynomials with
respect to L are related to primary fields in lattice theories.
