  Let E be an elliptic curve defined over the rationals and let N be its
conductor. Assume N is prime. In this paper we give numerical evidence that
suggests some conjectures on the 2-divisibility of certain sums of Heenger
points on E of discriminant D dividing 4N. One of these conjectures suggests a
possible link between the parity of the eigenvalue a_A(2) and the parity of the
Shafarevich-Tate group of certain elliptic curves A of square conductor.
