# List of coset representatives for $$\Gamma_H(N)$$ in $${\rm SL}_2(\ZZ)$$¶

class sage.modular.modsym.ghlist.GHlist(group)

A class representing a list of coset representatives for $$\Gamma_H(N)$$ in $${\rm SL}_2(\ZZ)$$.

TESTS:

sage: L = sage.modular.modsym.ghlist.GHlist(GammaH(18,[13]))
True

list()

Return a list of vectors representing the cosets. Do not change the returned list!

EXAMPLE:

sage: L = sage.modular.modsym.ghlist.GHlist(GammaH(4,[])); L.list()
[(0, 1), (0, 3), (1, 0), (1, 1), (1, 2), (1, 3), (2, 1), (2, 3), (3, 0), (3, 1), (3, 2), (3, 3)]

normalize(u, v)

Given a pair $$(u,v)$$ of integers, return the unique pair $$(u', v')$$ such that the pair $$(u', v')$$ appears in self.list() and $$(u, v)$$ is equivalent to $$(u', v')$$.

This will only make sense if $${\rm gcd}(u, v, N) = 1$$; otherwise the output will not be an element of self.

EXAMPLES:

sage: sage.modular.modsym.ghlist.GHlist(GammaH(24, [17, 19])).normalize(17, 6)
(1, 6)
sage: sage.modular.modsym.ghlist.GHlist(GammaH(24, [7, 13])).normalize(17, 6)
(5, 6)
sage: sage.modular.modsym.ghlist.GHlist(GammaH(24, [5, 23])).normalize(17, 6)
(7, 18)


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