  Let I be an arbitrary ideal in C[[x,y]]. We use the Newton algorithm to
compute by induction the motivic zeta function of the ideal, yielding only few
poles, associated to the faces of the successive Newton polygons. We associate
a minimal Newton tree to I, related to using good coordinates in the Newton
algorithm, and show that it has a conceptual geometric interpretation in terms
of the log canonical model of I. We also compute the log canonical threshold
from a Newton polygon and strengthen Corti's inequalities.
