  The leading order term for the average, over quadratic discriminants
satisfying the so-called Heegner condition, of the Neron-Tate height of Heegner
points on a rational elliptic curve E has been determined in [12]. In addition,
the second order term has been conjectured. In this paper, we prove that this
conjectured second order term is the right one; this yields a power saving in
the remainder term. Cancellations of Fourier coefficients of GL(2)-cusp forms
in arithmetic progressions lie in the core of the proof.
