Regular maps and principal congruence subgroups of Hecke groups

Regular q-valent maps correspond to normal subgroups of the triangle group (2,q,∞). This group has a representation as the Hecke group Hq which is generated by z→−1z and z→−1z+λq, where λq≔2cosπq. We investigate the regular maps corresponding to the principal congruence subgroups of Hq. Those of low index give many interesting regular maps.