  We study the asymptotic behavior of the heat trace coefficients $a_n$ as n
tends to infinity for the scalar Laplacian in the context of locally symmetric
spaces. We show that if the Plancherel measure of a noncompact type symmetric
space is polynomial, then these coefficients decay like 1/n!. On the other
hand, for even dimensional locally rank 1-symmetric spaces, one has $|a_n|$
grows like C^n* n! for some C>0; we conjecture this is the case in general if
the associated Plancherel measure is not polynomial. These examples show that
growth estimates conjectured by Berry and Howls are sharp. We also construct
examples of locally symmetric spaces which are not irreducible, which are not
flat, and so that only a finite number of the heat trace coefficients are
non-zero.
