The modular automorphisms of quotient modular curves

We obtain the modular automorphism group of any quotient modular curve of level , with . In particular, we obtain some unexpected automorphisms of order 3 that appear for the quotient modular curves when the Atkin–Lehner involution belongs to the quotient modular group. We also prove that such automorphisms are not necessarily defined over . As a consequence of these results, we obtain the full automorphism group of the quotient modular curve , for sufficiently large .