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Given a congruence subgroup, you can get another data structure for it given exactly by the action of the standard generators (L = parabolic fixing infinity, R = parbolic fixing 0, S2 = element of order 2, S3 = element of order 3). You may proceed as follows for the first example of the list

sage: G = Gamma(4)
sage: GG = G.as_permutation_group()  
sage: GG  
Arithmetic subgroup of index 48
sage: GG.L() 
sage: GG.R()
sage: GG.S2()
sage: GG.S3()

Warning: it is the right action on right coset (ie of the form Hg).

Note that you can also plot a fundamental domain using Farey symbols (Kulkarni method)

sage: G = Gamma(4)
sage: FareySymbol(G).fundamental_domain()