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We study “how far away” a finite index subgroup  Γ  of   ${\mathrm {SL}}_2({\mathbb Z})$   is from being a congruence group. For this, we define its  deficiency of being a congruence group . We show that the index of the image of  Γ  in   ${\mathrm {SL}}_2({\mathbb Z}/n{\mathbb Z})$   is the biggest, if  n  is the general Wohlfahrt level. We, furthermore, show that the Veech groups of origamis (or square-tiled surfaces) in   ${\mathcal H}_{2}(2)$   are far away from being congruence groups and that in each genus 1 finds an infinite family of origamis such that they are “as far as possible” from being a congruence group.

The Deficiency of being a Congruence Group for Veech Groups of Origamis