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# Congruence Subgroup \(\Gamma_0(N)\)

class `sage.modular.arithgroup.congroup_gamma0.``Gamma0_class`(level)

  Bases: `sage.modular.arithgroup.congroup_gammaH.GammaH_class`

  The congruence subgroup \(\Gamma_0(N)\).

  TESTS:

      sage: Gamma0(11).dimension_cusp_forms(2)
      1
      sage: a = Gamma0(1).dimension_cusp_forms(2); a
      0
      sage: type(a)
      <type 'sage.rings.integer.Integer'>
      sage: Gamma0(5).dimension_cusp_forms(0)
      0
      sage: Gamma0(20).dimension_cusp_forms(1)
      0
      sage: Gamma0(20).dimension_cusp_forms(4)
      6

      sage: Gamma0(23).dimension_cusp_forms(2)
      2
      sage: Gamma0(1).dimension_cusp_forms(24)
      2
      sage: Gamma0(3).dimension_cusp_forms(3)
      0
      sage: Gamma0(11).dimension_cusp_forms(-1)
      0

      sage: Gamma0(22).dimension_new_cusp_forms()
      0
      sage: Gamma0(100).dimension_new_cusp_forms(2, 5)
      5

  Independently compute the dimension 5 above:

      sage: m = ModularSymbols(100, 2,sign=1).cuspidal_subspace()
      sage: m.new_subspace(5)
      Modular Symbols subspace of dimension 5 of Modular Symbols space of dimension 18 for Gamma_0(100) of weight 2 with sign 1 over Rational Field

  `coset_reps`()

    Return representatives for the right cosets of this congruence subgroup in \({\rm SL}_2(\ZZ)\) as a generator object.

    Use `list(self.coset_reps())` to obtain coset reps as a list.

    EXAMPLES:

        sage: list(Gamma0(5).coset_reps())
        [
        [1 0]  [ 0 -1]  [1 0]  [ 0 -1]  [ 0 -1]  [ 0 -1]
        [0 1], [ 1  0], [1 1], [ 1  2], [ 1  3], [ 1  4]
        ]
        sage: list(Gamma0(4).coset_reps())
        [
        [1 0]  [ 0 -1]  [1 0]  [ 0 -1]  [ 0 -1]  [1 0]
        [0 1], [ 1  0], [1 1], [ 1  2], [ 1  3], [2 1]
        ]
        sage: list(Gamma0(1).coset_reps())
        [
        [1 0]
        [0 1]
        ]

  `divisor_subgroups`()

    Return the subgroups of SL2Z of the form Gamma0(M) that contain this subgroup, i.e. those for M a divisor of N.

    EXAMPLE:

        sage: Gamma0(24).divisor_subgroups()
        [Modular Group SL(2,Z),
        Congruence Subgroup Gamma0(2),
        Congruence Subgroup Gamma0(3),
        Congruence Subgroup Gamma0(4),
        Congruence Subgroup Gamma0(6),
        Congruence Subgroup Gamma0(8),
        Congruence Subgroup Gamma0(12),
        Congruence Subgroup Gamma0(24)]

  `gamma_h_subgroups`()

    Return the subgroups of the form \(\Gamma_H(N)\) contained in self, where \(N\) is the level of self.

    EXAMPLES:

        sage: G = Gamma0(11)
        sage: G.gamma_h_subgroups()
        [Congruence Subgroup Gamma0(11), Congruence Subgroup Gamma_H(11) with H generated by [4], Congruence Subgroup Gamma_H(11) with H generated by [10], Congruence Subgroup Gamma1(11)]
        sage: G = Gamma0(12)
        sage: G.gamma_h_subgroups()
        [Congruence Subgroup Gamma0(12), Congruence Subgroup Gamma_H(12) with H generated by [7], Congruence Subgroup Gamma_H(12) with H generated by [11], Congruence Subgroup Gamma_H(12) with H generated by [5], Congruence Subgroup Gamma1(12)]

  `generators`(algorithm='farey')

    Return generators for this congruence subgroup.

    INPUT:

    - `algorithm` (string): either `farey` (default) or `todd-coxeter`.

    If `algorithm` is set to `"farey"`, then the generators will be calculated using Farey symbols, which will always return a minimal generating set. See `farey_symbol` for more information.

    If `algorithm` is set to `"todd-coxeter"`, a simpler algorithm based on Todd-Coxeter enumeration will be used. This tends to return far larger sets of generators.

    EXAMPLE:

        sage: Gamma0(3).generators()
        [
        [1 1]  [-1  1]
        [0 1], [-3  2]
        ]
        sage: Gamma0(3).generators(algorithm="todd-coxeter")
        [
        [1 1]  [-1  0]  [ 1 -1]  [1 0]  [1 1]  [-1  0]  [ 1  0]
        [0 1], [ 0 -1], [ 0  1], [3 1], [0 1], [ 3 -1], [-3  1]
        ]
        sage: SL2Z.gens()
        (
        [ 0 -1]  [1 1]
        [ 1  0], [0 1]
        )

  `index`()

    Return the index of self in the full modular group. This is given by

    \[\begin{split}N \prod_{\substack{p \mid N \\ \text{$p$ prime}}}\left(1 + \frac{1}{p}\right).\end{split}\]
    EXAMPLE::
      sage: [Gamma0(n).index() for n in [1..19]] [1, 3, 4, 6, 6, 12, 8, 12, 12, 18, 12, 24, 14, 24, 24, 24, 18, 36, 20] sage: Gamma0(32041).index() 32220
  `is_even`()

    Return True precisely if this subgroup contains the matrix -1.

    Since \(\Gamma0(N)\) always contains the matrix -1, this always returns True.

    EXAMPLES:

        sage: Gamma0(12).is_even()
        True
        sage: SL2Z.is_even()
        True

  `is_subgroup`(right)

    Return True if self is a subgroup of right.

    EXAMPLES:

        sage: G = Gamma0(20)
        sage: G.is_subgroup(SL2Z)
        True
        sage: G.is_subgroup(Gamma0(4))
        True
        sage: G.is_subgroup(Gamma0(20))
        True
        sage: G.is_subgroup(Gamma0(7))
        False
        sage: G.is_subgroup(Gamma1(20))
        False
        sage: G.is_subgroup(GammaH(40, []))
        False
        sage: Gamma0(80).is_subgroup(GammaH(40, [31, 21, 17]))
        True
        sage: Gamma0(2).is_subgroup(Gamma1(2))
        True

  `ncusps`()

    Return the number of cusps of this subgroup \(\Gamma_0(N)\).

    EXAMPLES:

        sage: [Gamma0(n).ncusps() for n in [1..19]]
        [1, 2, 2, 3, 2, 4, 2, 4, 4, 4, 2, 6, 2, 4, 4, 6, 2, 8, 2]
        sage: [Gamma0(n).ncusps() for n in prime_range(2,100)]
        [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]

  `nu2`()

    Return the number of elliptic points of order 2 for this congruence subgroup \(\Gamma_0(N)\). The number of these is given by a standard formula: 0 if \(N\) is divisible by 4 or any prime congruent to -1 mod 4, and otherwise \(2^d\) where d is the number of odd primes dividing \(N\).

    EXAMPLE:

        sage: Gamma0(2).nu2()
        1
        sage: Gamma0(4).nu2()
        0
        sage: Gamma0(21).nu2()
        0
        sage: Gamma0(1105).nu2()
        8
        sage: [Gamma0(n).nu2() for n in [1..19]]
        [1, 1, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 2, 0, 0, 0, 2, 0, 0]

  `nu3`()

    Return the number of elliptic points of order 3 for this congruence subgroup \(\Gamma_0(N)\). The number of these is given by a standard formula: 0 if \(N\) is divisible by 9 or any prime congruent to -1 mod 3, and otherwise \(2^d\) where d is the number of primes other than 3 dividing \(N\).

    EXAMPLE:

        sage: Gamma0(2).nu3()
        0
        sage: Gamma0(3).nu3()
        1
        sage: Gamma0(9).nu3()
        0
        sage: Gamma0(7).nu3()
        2
        sage: Gamma0(21).nu3()
        2
        sage: Gamma0(1729).nu3()
        8

`sage.modular.arithgroup.congroup_gamma0.``Gamma0_constructor`(N)

  Return the congruence subgroup Gamma0(N).

  EXAMPLES:

      sage: G = Gamma0(51) ; G # indirect doctest
      Congruence Subgroup Gamma0(51)
      sage: G == Gamma0(51)
      True
      sage: G is Gamma0(51)
      True

`sage.modular.arithgroup.congroup_gamma0.``is_Gamma0`(x)

  Return True if x is a congruence subgroup of type Gamma0.

  EXAMPLES:

      sage: from sage.modular.arithgroup.all import is_Gamma0
      sage: is_Gamma0(SL2Z)
      True
      sage: is_Gamma0(Gamma0(13))
      True
      sage: is_Gamma0(Gamma1(6))
      False

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