A Mock Theta Function for the Delta-function

We define a "mock theta" function M(Delta)(z) = (n=-1)Sigma(infinity) a(Delta)(n)q(n) = 11! . q(-1) - 2615348736000/691 - ... for Ramanujan's Delta-function. Although it is not a modular form, we are able to determine its images under the Hecke operators. These images are in terms of Ramanujan's tau-function and the modular polynomials j(m)(z) which encode the denominator formula for the infinite dimensional Monster Lie algebra. Using these results, we obtain a criterion for Lehmer's Conjecture on the nonvanishing of tau(n). We show that tau(p) = 0 if and only if M(Delta)(z) | T(-10)(p) is a modular form. We also relate Lehmer's Conjecture to integrality properties of the a(Delta)(n).