Weierstrass points on certain modular groups

We investigate Weierstrass points of the modular curve XΔ(N)XΔ(N) of genus ≥2 when Δ is a proper subgroup of (Z/NZ)⁎(Z/NZ)⁎. Let N=p2MN=p2M where p is a prime number and M   is a positive integer. Modifying Atkin's method in the case ±(1+pM)∈Δ±(1+pM)∈Δ, we find conditions for the cusp 0 to be a Weierstrass point on the modular curve XΔ(p2M)XΔ(p2M). Moreover, applying Schöneberg's theorem we show that except for finitely many N  , the fixed points of the Fricke involutions WNWN are Weierstrass points on XΔ(N)XΔ(N).