  ATR points were introduced by Darmon as a conjectural construction of
algebraic points on certain elliptic curves for which in general the Heegner
point method is not available. So far the only numerical evidence, provided by
Darmon--Logan and G\"artner, concerned curves arising as quotients of Shimura
curves. In those special cases the ATR points can be obtained from the already
existing Heegner points, thanks to results of Zhang and Darmon--Rotger--Zhao.
  In this paper we compute for the first time an algebraic ATR point on a curve
which is not uniformizable by any Shimura curve, thus providing the first piece
of numerical evidence that Darmon's construction works beyond geometric
modularity. To this purpose we improve the method proposed by Darmon and Logan
by removing the requirement that the real quadratic field be norm-euclidean,
and accelerating the numerical integration of Hilbert modular forms.
