congroup_gammaH.py

r"""
Congruence Subgroup `\Gamma_H(N)`

AUTHORS:

- Jordi Quer
- David Loeffler
"""

################################################################################
#
#       Copyright (C) 2009, The Sage Group -- http://www.sagemath.org/
#
#  Distributed under the terms of the GNU General Public License (GPL)
#
#  The full text of the GPL is available at:
#
#                  http://www.gnu.org/licenses/
#
################################################################################

from sage.rings.arith import euler_phi, lcm, gcd, divisors, get_inverse_mod, get_gcd, factor
from sage.modular.modsym.p1list import lift_to_sl2z
from congroup_generic import CongruenceSubgroup, is_CongruenceSubgroup
from arithgroup_element import ArithmeticSubgroupElement
from sage.modular.cusps import Cusp
from sage.misc.cachefunc import cached_method

# Just for now until we make an SL_2 group type.
from sage.rings.integer_ring import ZZ
from sage.rings.finite_rings.integer_mod_ring import Zmod
from sage.matrix.matrix_space import MatrixSpace
from sage.groups.matrix_gps.matrix_group import MatrixGroup
from sage.matrix.constructor import matrix

Mat2Z = MatrixSpace(ZZ,2)


_gammaH_cache = {}
def GammaH_constructor(level, H):
    r"""
    Return the congruence subgroup `\Gamma_H(N)`, which is the subgroup of
    `SL_2(\ZZ)` consisting of matrices of the form `\begin{pmatrix} a & b \\
    c & d \end{pmatrix}` with `N | c` and `a, b \in H`, for `H` a specified
    subgroup of `(\ZZ/N\ZZ)^\times`.
    
    INPUT:
    
    - level -- an integer
    - H -- either 0, 1, or a list
        * If H is a list, return `\Gamma_H(N)`, where `H`
          is the subgroup of `(\ZZ/N\ZZ)^*` **generated** by the
          elements of the list. 
        * If H = 0, returns `\Gamma_0(N)`.
        * If H = 1, returns `\Gamma_1(N)`.

    EXAMPLES::

        sage: GammaH(11,0) # indirect doctest
        Congruence Subgroup Gamma0(11)
        sage: GammaH(11,1)
        Congruence Subgroup Gamma1(11)
        sage: GammaH(11,[10])
        Congruence Subgroup Gamma_H(11) with H generated by [10]
        sage: GammaH(11,[10,1])
        Congruence Subgroup Gamma_H(11) with H generated by [10]
        sage: GammaH(14,[10])
        Traceback (most recent call last):
        ...
        ArithmeticError: The generators [10] must be units modulo 14
    """
    from all import Gamma0, Gamma1, SL2Z
    if level == 1: 
        return SL2Z
    elif H == 0:
        return Gamma0(level)
    elif H == 1:
        return Gamma1(level)

    H = _normalize_H(H, level)
    if H == []:
        return Gamma1(level)

    Hlist = _list_subgroup(level, H)
    if len(Hlist) == euler_phi(level):
        return Gamma0(level)

    key = (level, tuple(H))
    try:
        return _gammaH_cache[key]
    except KeyError:
        _gammaH_cache[key] = GammaH_class(level, H, Hlist)
        return _gammaH_cache[key]
    
def is_GammaH(x):
    """
    Return True if x is a congruence subgroup of type GammaH.

    EXAMPLES::

        sage: from sage.modular.arithgroup.all import is_GammaH
        sage: is_GammaH(GammaH(13, [2]))
        True
        sage: is_GammaH(Gamma0(6))
        True
        sage: is_GammaH(Gamma1(6))
        True
        sage: is_GammaH(sage.modular.arithgroup.congroup_generic.CongruenceSubgroup(5))
        False
    """
    return isinstance(x, GammaH_class)

def _normalize_H(H, level):
    """
    Normalize representatives for a given subgroup H of the units
    modulo level.

    NOTE: This function does *not* make any attempt to find a minimal
    set of generators for H. It simply normalizes the inputs for use
    in hashing.

    EXAMPLES::

        sage: sage.modular.arithgroup.congroup_gammaH._normalize_H([23], 10)
        [3]
        sage: sage.modular.arithgroup.congroup_gammaH._normalize_H([1,5], 7)
        [5]
        sage: sage.modular.arithgroup.congroup_gammaH._normalize_H([4,18], 14)
        Traceback (most recent call last):
        ...
        ArithmeticError: The generators [4, 18] must be units modulo 14
        sage: sage.modular.arithgroup.congroup_gammaH._normalize_H([3,17], 14)
        [3]
        sage: sage.modular.arithgroup.congroup_gammaH._normalize_H([-1,7,9], 10)
        [7, 9]
    """
    if not isinstance(H, list):
        raise TypeError, "H must be a list."
    H = [ZZ(h) for h in H]
    for h in H:
        if gcd(h, level) > 1:
            raise ArithmeticError, 'The generators %s must be units modulo %s'%(H, level)
    H = list(set([h%level for h in H]))
    H.sort()
    if 1 in H:
        H.remove(1)
    return H

class GammaH_class(CongruenceSubgroup):

    r"""
    The congruence subgroup `\Gamma_H(N)` for some subgroup `H \trianglelefteq
    (\ZZ / N\ZZ)^\times`, which is the subgroup of `{\rm
    SL}_2(\ZZ)` consisting of matrices of the form `\begin{pmatrix} a &
    b \\ c & d \end{pmatrix}` with `N \mid c` and `a, b \in H`.

    TESTS:

    We test calculation of various invariants of the group: ::
        
        sage: GammaH(33,[2]).projective_index()
        96
        sage: GammaH(33,[2]).genus()
        5
        sage: GammaH(7,[2]).genus()
        0
        sage: GammaH(23, [1..22]).genus()
        2
        sage: Gamma0(23).genus()
        2
        sage: GammaH(23, [1]).genus()
        12
        sage: Gamma1(23).genus()
        12

    We calculate the dimensions of some modular forms spaces: ::

        sage: GammaH(33,[2]).dimension_cusp_forms(2)
        5
        sage: GammaH(33,[2]).dimension_cusp_forms(3)
        0
        sage: GammaH(33,[2,5]).dimension_cusp_forms(2)
        3
        sage: GammaH(32079, [21676]).dimension_cusp_forms(20)
        180266112

    We can sometimes show that there are no weight 1 cusp forms: ::

        sage: GammaH(20, [9]).dimension_cusp_forms(1)
        0

    """

    def __init__(self, level, H, Hlist=None):
        r"""
        The congruence subgroup `\Gamma_H(N)`. The subgroup H
        must be input as a list.

        EXAMPLES::

            sage: GammaH(117, [4]) 
            Congruence Subgroup Gamma_H(117) with H generated by [4]
            sage: G = GammaH(16, [7])
            sage: G == loads(dumps(G))
            True
            sage: G is loads(dumps(G))
            True
        """
        CongruenceSubgroup.__init__(self, level)
        self.__H = H
        if Hlist is None: Hlist = _list_subgroup(level, H)
        self.__Hlist = Hlist

    def restrict(self, M):
        r"""
        Return the subgroup of `\Gamma_0(M)`, for `M` a divisor of `N`,
        obtained by taking the image of this group under reduction modulo `N`.
        
        EXAMPLES::

            sage: G = GammaH(33,[2])
            sage: G.restrict(11)
            Congruence Subgroup Gamma0(11)
            sage: G.restrict(1)
            Modular Group SL(2,Z)
            sage: G.restrict(15)
            Traceback (most recent call last):
            ...
            ValueError: M (=15) must be a divisor of the level (33) of self
        """
        M = ZZ(M)
        if self.level() % M:
            raise ValueError, "M (=%s) must be a divisor of the level (%s) of self" % (M, self.level())
        return self._new_group_from_level(M)

    def extend(self, M):
        r"""
        Return the subgroup of `\Gamma_0(M)`, for `M` a multiple of `N`,
        obtained by taking the preimage of this group under the reduction map;
        in other words, the intersection of this group with `\Gamma_0(M)`.
        
        EXAMPLES::

            sage: G = GammaH(33, [2])
            sage: G.extend(99)
            Congruence Subgroup Gamma_H(99) with H generated by [2, 35, 68]
            sage: G.extend(11)
            Traceback (most recent call last):
            ...
            ValueError: M (=11) must be a multiple of the level (33) of self

        """
        M = ZZ(M)
        if M % self.level():
            raise ValueError, "M (=%s) must be a multiple of the level (%s) of self" % (M, self.level())
        return self._new_group_from_level(M)

    def __reduce__(self):
        """
        Used for pickling self.

        EXAMPLES::

            sage: GammaH(92,[45,47]).__reduce__()
            (<function GammaH_constructor at ...>, (92, [45, 47]))
        """
        return GammaH_constructor, (self.level(), self.__H)

    def divisor_subgroups(self):
        r"""
        Given this congruence subgroup `\Gamma_H(N)`, return all
        subgroups `\Gamma_G(M)` for `M` a divisor of `N` and such that
        `G` is equal to the image of `H` modulo `M`.

        EXAMPLES::

            sage: G = GammaH(33,[2]); G
            Congruence Subgroup Gamma_H(33) with H generated by [2]
            sage: G._list_of_elements_in_H()
            [1, 2, 4, 8, 16, 17, 25, 29, 31, 32]
            sage: G.divisor_subgroups()
            [Modular Group SL(2,Z), 
             Congruence Subgroup Gamma0(3),
             Congruence Subgroup Gamma0(11),
             Congruence Subgroup Gamma_H(33) with H generated by [2]]
        """
        v = self.__H
        ans = []
        for M in self.level().divisors():
            w = [a % M for a in v if a%M]
            ans.append(GammaH_constructor(M, w))
        return ans

    def to_even_subgroup(self):
        r""" 
        Return the smallest even subgroup of `SL(2, \ZZ)` containing self.

        EXAMPLE::

            sage: GammaH(11, [4]).to_even_subgroup()
            Congruence Subgroup Gamma0(11)
            sage: Gamma1(11).to_even_subgroup()
            Congruence Subgroup Gamma_H(11) with H generated by [10]

        """
        if self.is_even(): return self
        else:
            return GammaH_constructor(self.level(), self._generators_for_H() + [-1])

    def __cmp__(self, other):
        """
        Compare self to other.

        The ordering on congruence subgroups of the form GammaH(N) for
        some H is first by level and then by the subgroup H. In
        particular, this means that we have Gamma1(N) < GammaH(N) <
        Gamma0(N) for every nontrivial subgroup H.

        EXAMPLES::

            sage: G = GammaH(86, [9])
            sage: G.__cmp__(G)
            0
            sage: G.__cmp__(GammaH(86, [11])) is not 0
            True
            sage: Gamma1(11) < Gamma0(11)
            True
            sage: Gamma1(11) == GammaH(11, [])
            True
            sage: Gamma0(11) == GammaH(11, [2])
            True
        """
        if is_GammaH(other):
            t = cmp(self.level(), other.level())
            if t: 
                return t
            else: 
                return cmp(self._list_of_elements_in_H(), other._list_of_elements_in_H())
        else:
            return CongruenceSubgroup.__cmp__(self, other)

    def _generators_for_H(self):
        """
        Return generators for the subgroup H of the units mod
        self.level() that defines self.
        
        EXAMPLES::

            sage: GammaH(17,[4])._generators_for_H()
            [4]
            sage: GammaH(12,[-1])._generators_for_H()
            [11]
        """
        return self.__H

    def _repr_(self):
        """
        Return the string representation of self.

        EXAMPLES::

            sage: GammaH(123, [55])._repr_()
            'Congruence Subgroup Gamma_H(123) with H generated by [55]'
        """
        return "Congruence Subgroup Gamma_H(%s) with H generated by %s"%(self.level(), self.__H)

    def _latex_(self):
        r"""
        Return the \LaTeX representation of self.
        
        EXAMPLES::

            sage: GammaH(5,[4])._latex_()
            '\\Gamma_H(5)'
        """
        return "\\Gamma_H(%s)"%self.level()

    def _list_of_elements_in_H(self):
        """
        Returns a sorted list of Python ints that are representatives
        between 1 and N-1 of the elements of H.

        WARNING: Do not change this returned list. 

        EXAMPLES::

            sage: G = GammaH(11,[3]); G
            Congruence Subgroup Gamma_H(11) with H generated by [3]
            sage: G._list_of_elements_in_H()
            [1, 3, 4, 5, 9]        
        """
        return self.__Hlist

    def is_even(self):
        """
        Return True precisely if this subgroup contains the matrix -1.

        EXAMPLES::

            sage: GammaH(10, [3]).is_even()
            True
            sage: GammaH(14, [1]).is_even()
            False
        """
        if self.level() == 1:
            return True
        v = self._list_of_elements_in_H()
        return int(self.level() - 1) in v

    @cached_method
    def generators(self, algorithm="farey"):
        r"""
        Return generators for this congruence subgroup. The result is cached.
 
        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 :mod:`~sage.modular.arithgroup.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: GammaH(7, [2]).generators()
            [
            [1 1]  [ 2 -1]  [ 4 -3]
            [0 1], [ 7 -3], [ 7 -5]
            ]
            sage: GammaH(7, [2]).generators(algorithm="todd-coxeter")
            [
            [1 1]  [-90  29]  [ 15   4]  [-10  -3]  [ 1 -1]  [1 0]  [1 1]  [-3 -1]
            [0 1], [301 -97], [-49 -13], [  7   2], [ 0  1], [7 1], [0 1], [ 7  2],
            <BLANKLINE>
            [-13   4]  [-5 -1]  [-5 -2]  [-10   3]  [ 1  0]  [ 9 -1]  [-20   7]
            [ 42 -13], [21  4], [28 11], [ 63 -19], [-7  1], [28 -3], [-63  22],
            <BLANKLINE>
            [1 0]  [-3 -1]  [ 15  -4]  [ 2 -1]  [ 22  -7]  [-5  1]  [  8  -3]
            [7 1], [ 7  2], [ 49 -13], [ 7 -3], [ 63 -20], [14 -3], [-21   8],
            <BLANKLINE>
            [11  5]  [-13  -4]
            [35 16], [-42 -13]
            ]
        """
        if algorithm=="farey":
            return self.farey_symbol().generators()
        elif algorithm=="todd-coxeter":
            from sage.modular.modsym.ghlist import GHlist
            from congroup_pyx import generators_helper
            level = self.level()
            gen_list = generators_helper(GHlist(self), level, Mat2Z)
            return [self(g, check=False) for g in gen_list]
        else:
            raise ValueError, "Unknown algorithm '%s' (should be either 'farey' or 'todd-coxeter')" % algorithm
    
    def _coset_reduction_data_first_coord(G):
        """
        Compute data used for determining the canonical coset
        representative of an element of SL_2(Z) modulo G. This
        function specifically returns data needed for the first part
        of the reduction step (the first coordinate).

        INPUT:
            G -- a congruence subgroup Gamma_0(N), Gamma_1(N), or Gamma_H(N).

        OUTPUT:
            A list v such that 
                v[u] = (min(u*h: h in H),
                        gcd(u,N) ,
                        an h such that h*u = min(u*h: h in H)).

        EXAMPLES::

            sage: G = Gamma0(12)
            sage: sage.modular.arithgroup.congroup_gammaH.GammaH_class._coset_reduction_data_first_coord(G)
            [(0, 12, 0), (1, 1, 1), (2, 2, 1), (3, 3, 1), (4, 4, 1), (1, 1, 5), (6, 6, 1),
            (1, 1, 7), (4, 4, 5), (3, 3, 7), (2, 2, 5), (1, 1, 11)]
        """
        H = [ int(x) for x in G._list_of_elements_in_H() ]
        N = int(G.level())

        # Get some useful fast functions for inverse and gcd
        inverse_mod = get_inverse_mod(N)   # optimal inverse function
        gcd = get_gcd(N)   # optimal gcd function

        # We will be filling this list in below.
        reduct_data = [0] * N

        # We can fill in 0 and all elements of H immediately
        reduct_data[0] = (0,N,0)
        for u in H:
            reduct_data[u] = (1, 1, inverse_mod(u, N))
        
        # Make a table of the reduction of H (mod N/d), one for each
        # divisor d.
        repr_H_mod_N_over_d = {}    
        for d in divisors(N):
            # We special-case N == d because in this case,
            # 1 % N_over_d is 0
            if N == d:
                repr_H_mod_N_over_d[d] = [1]
                break
            N_over_d = N//d
            # For each element of H, we look at its image mod
            # N_over_d. If we haven't yet seen it, add it on to
            # the end of z.
            w = [0] * N_over_d
            z = [1]
            for x in H:
                val = x%N_over_d
                if not w[val]:
                    w[val] = 1
                    z.append(x)
            repr_H_mod_N_over_d[d] = z

        # Compute the rest of the tuples. The values left to process
        # are those where reduct_data has a 0. Note that several of
        # these values are processed on each loop below, so re-index
        # each time.
        while True:
            try:
                u = reduct_data.index(0)
            except ValueError:
                break
            d = gcd(u, N)
            for x in repr_H_mod_N_over_d[d]:
                reduct_data[(u*x)%N] = (u, d, inverse_mod(x,N))

        return reduct_data

    def _coset_reduction_data_second_coord(G):
        """
        Compute data used for determining the canonical coset
        representative of an element of SL_2(Z) modulo G. This
        function specifically returns data needed for the second part
        of the reduction step (the second coordinate).

        INPUT:
            self

        OUTPUT:
            a dictionary v with keys the divisors of N such that v[d]
            is the subgroup {h in H : h = 1 (mod N/d)}.

        EXAMPLES::

            sage: G = GammaH(240,[7,239])
            sage: G._coset_reduction_data_second_coord()
            {1: [1], 2: [1], 3: [1], 4: [1], 5: [1, 49], 6: [1], 48: [1, 191], 8: [1], 80: [1, 7, 49, 103], 10: [1, 49], 12: [1], 15: [1, 49], 240: [1, 7, 49, 103, 137, 191, 233, 239], 40: [1, 7, 49, 103], 20: [1, 49], 24: [1, 191], 120: [1, 7, 49, 103, 137, 191, 233, 239], 60: [1, 49, 137, 233], 30: [1, 49, 137, 233], 16: [1]}
            sage: G = GammaH(1200,[-1,7]); G
            Congruence Subgroup Gamma_H(1200) with H generated by [7, 1199]
            sage: K = G._coset_reduction_data_second_coord().keys() ; K.sort()
            sage: K == divisors(1200)
            True
        """
        H = G._list_of_elements_in_H()
        N = G.level()
        v = { 1: [1] , N: H }
        for d in [x for x in divisors(N) if x > 1 and x < N ]: 
            N_over_d = N // d
            v[d] = [x for x in H if x % N_over_d == 1]
        return v        

    @cached_method
    def _coset_reduction_data(self):
        """
        Compute data used for determining the canonical coset
        representative of an element of SL_2(Z) modulo G.

        EXAMPLES::

            sage: G = GammaH(13, [-1]); G
            Congruence Subgroup Gamma_H(13) with H generated by [12]
            sage: G._coset_reduction_data()
            ([(0, 13, 0), (1, 1, 1), (2, 1, 1), (3, 1, 1), (4, 1, 1), (5, 1, 1), (6, 1, 1), (6, 1, 12), (5, 1, 12), (4, 1, 12), (3, 1, 12), (2, 1, 12), (1, 1, 12)], {1: [1], 13: [1, 12]})
        """
        return (self._coset_reduction_data_first_coord(),
                                       self._coset_reduction_data_second_coord())


    def _reduce_coset(self, uu, vv):
        r"""
        Compute a canonical form for a given Manin symbol.
        
        INPUT:
        Two integers (uu,vv) that define an element of `(Z/NZ)^2`.
            uu -- an integer
            vv -- an integer

        OUTPUT:
           pair of integers that are equivalent to (uu,vv).

        NOTE: We do *not* require that gcd(uu,vv,N) = 1.  If the gcd is
        not 1, we return (0,0).

        EXAMPLES:
        
        An example at level 9.::

            sage: G = GammaH(9,[7]); G
            Congruence Subgroup Gamma_H(9) with H generated by [7]
            sage: a = []
            sage: for i in range(G.level()):
            ...     for j in range(G.level()):
            ...       a.append(G._reduce_coset(i,j))
            sage: v = list(set(a))
            sage: v.sort()
            sage: v
            [(0, 0), (0, 1), (0, 2), (1, 0), (1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), (1, 7), (1, 8), (2, 0), (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6), (2, 7), (2, 8), (3, 1), (3, 2), (6, 1), (6, 2)]

        An example at level 100::

            sage: G = GammaH(100,[3,7]); G
            Congruence Subgroup Gamma_H(100) with H generated by [3, 7]
            sage: a = []
            sage: for i in range(G.level()):
            ...   for j in range(G.level()):
            ...       a.append(G._reduce_coset(i,j))
            sage: v = list(set(a))
            sage: v.sort()
            sage: len(v)
            361

        This demonstrates the problem underlying trac \#1220::

            sage: G = GammaH(99, [67])
            sage: G._reduce_coset(11,-3)
            (11, 96)
            sage: G._reduce_coset(77, -3)
            (11, 96)
        """
        N = int(self.level())
        u = uu % N
        v = vv % N
        first, second = self._coset_reduction_data()

        if gcd(first[u][1], first[v][1]) != 1:
            return (0,0)
        if not u:
            return (0, first[v][0])
        if not v:
            return (first[u][0], 0)

        new_u = first[u][0]
        d = first[u][1]
        new_v = (first[u][2] * v) % N
        H_ls = second[d]
        if len(H_ls) > 1:
            new_v = min([ (new_v * h)%N for h in H_ls ])

        return (new_u, new_v)

    def reduce_cusp(self, c):
        r"""
        Compute a minimal representative for the given cusp c. Returns
        a cusp c' which is equivalent to the given cusp, and is in
        lowest terms with minimal positive denominator, and minimal
        positive numerator for that denominator.
        
        Two cusps `u_1/v_1` and `u_2/v_2` are equivalent modulo `\Gamma_H(N)`
        if and only if

        .. math::

            v_1 =  h v_2 \bmod N\quad \text{and}\quad u_1 =  h^{-1} u_2 \bmod {\rm gcd}(v_1,N)
        
        or
        
        .. math::

            v_1 = -h v_2 \bmod N\quad \text{and}\quad u_1 = -h^{-1} u_2 \bmod {\rm gcd}(v_1,N)

        for some `h \in H`. 

        EXAMPLES::

            sage: GammaH(6,[5]).reduce_cusp(5/3)
            1/3
            sage: GammaH(12,[5]).reduce_cusp(Cusp(8,9))
            1/3
            sage: GammaH(12,[5]).reduce_cusp(5/12)
            Infinity
            sage: GammaH(12,[]).reduce_cusp(Cusp(5,12))
            5/12
            sage: GammaH(21,[5]).reduce_cusp(Cusp(-9/14))
            1/7
            sage: Gamma1(5).reduce_cusp(oo)
            Infinity
            sage: Gamma1(5).reduce_cusp(0)
            0
        """
        return self._reduce_cusp(c)[0]

    def _reduce_cusp(self, c):
        r"""
        Compute a minimal representative for the given cusp c. 
        Returns a pair (c', t), where c' is the minimal representative 
        for the given cusp, and t is either 1 or -1, as explained 
        below. Largely for internal use.
        
        The minimal representative for a cusp is the element in `P^1(Q)`
        in lowest terms with minimal positive denominator, and minimal
        positive numerator for that denominator.

        Two cusps `u1/v1` and `u2/v2` are equivalent modulo `\Gamma_H(N)`
        if and only if
            `v1 =  h*v2 (mod N)` and `u1 =  h^(-1)*u2 (mod gcd(v1,N))`
        or
            `v1 = -h*v2 (mod N)` and `u1 = -h^(-1)*u2 (mod gcd(v1,N))`
        for some `h \in H`. Then t is 1 or -1 as c and c' fall into
        the first or second case, respectively.

        EXAMPLES::

            sage: GammaH(6,[5])._reduce_cusp(Cusp(5,3))
            (1/3, -1)
            sage: GammaH(12,[5])._reduce_cusp(Cusp(8,9))
            (1/3, -1)
            sage: GammaH(12,[5])._reduce_cusp(Cusp(5,12))
            (Infinity, 1)
            sage: GammaH(12,[])._reduce_cusp(Cusp(5,12))
            (5/12, 1)
            sage: GammaH(21,[5])._reduce_cusp(Cusp(-9/14))
            (1/7, 1)
        """
        c = Cusp(c)
        N = int(self.level())
        Cusps = c.parent()
        v = int(c.denominator() % N)
        H = self._list_of_elements_in_H()

        # First, if N | v, take care of this case. If u is in \pm H,
        # then we return Infinity. If not, let u_0 be the minimum
        # of \{ h*u | h \in \pm H \}. Then return u_0/N.
        if not v:
            u = c.numerator() % N
            if u in H:
                return Cusps((1,0)), 1
            if (N-u) in H:
                return Cusps((1,0)), -1
            ls = [ (u*h)%N for h in H ]
            m1 = min(ls)
            m2 = N-max(ls)
            if m1 < m2:
                return Cusps((m1,N)), 1
            else:
                return Cusps((m2,N)), -1
        
        u = int(c.numerator() % v)
        gcd = get_gcd(N)
        d = gcd(v,N)

        # If (N,v) == 1, let v_0 be the minimal element
        # in \{ v * h | h \in \pm H \}. Then we either return
        # Infinity or 1/v_0, as v is or is not in \pm H,
        # respectively.
        if d == 1:
            if v in H:
                return Cusps((0,1)), 1
            if (N-v) in H:
                return Cusps((0,1)), -1
            ls = [ (v*h)%N for h in H ]
            m1 = min(ls)
            m2 = N-max(ls)
            if m1 < m2:
                return Cusps((1,m1)), 1
            else:
                return Cusps((1,m2)), -1

        val_min = v
        inv_mod = get_inverse_mod(N)

        # Now we're in the case (N,v) > 1. So we have to do several
        # steps: first, compute v_0 as above. While computing this
        # minimum, keep track of *all* pairs of (h,s) which give this
        # value of v_0.
        hs_ls = [(1,1)]
        for h in H:
            tmp = (v*h)%N
            
            if tmp < val_min:
                val_min = tmp
                hs_ls = [(inv_mod(h,N), 1)]
            elif tmp == val_min:
                hs_ls.append((inv_mod(h,N), 1))
                
            if (N-tmp) < val_min:
                val_min = N - tmp
                hs_ls = [(inv_mod(h,N), -1)]
            elif (N-tmp) == val_min:
                hs_ls.append((inv_mod(h,N), -1))

        # Finally, we find our minimal numerator. Let u_1 be the
        # minimum of s*h^-1*u mod d as (h,s) ranges over the elements
        # of hs_ls. We must find the smallest integer u_0 which is
        # smaller than v_0, congruent to u_1 mod d, and coprime to
        # v_0. Then u_0/v_0 is our minimal representative.
        u_min = val_min
        sign = None
        for h_inv,s in hs_ls:
            tmp = (h_inv * s * u)%d
            while gcd(tmp, val_min) > 1 and tmp < u_min:
                tmp += d
            if tmp < u_min:
                u_min = tmp
                sign = s

        return Cusps((u_min, val_min)), sign
     
    def _find_cusps(self):
        r"""
        Return an ordered list of inequivalent cusps for self, i.e. a
        set of representatives for the orbits of self on
        `\mathbf{P}^1(\QQ)`.  These are returned in a reduced
        form; see self.reduce_cusp for the definition of reduced.
        
        ALGORITHM:
            Lemma 3.2 in Cremona's 1997 book shows that for the action
            of Gamma1(N) on "signed projective space"
            `\Q^2 / (\Q_{\geq 0}^+)`, we have `u_1/v_1 \sim u_2 / v_2`
            if and only if `v_1 = v_2 \bmod N` and `u_1 = u_2 \bmod
            gcd(v_1, N)`. It follows that every orbit has a
            representative `u/v` with `v \le N` and `0 \le u \le
            gcd(v, N)`.  We iterate through all pairs `(u,v)`
            satisfying this.
            
            Having found a set containing at least one of every
            equivalence class modulo Gamma1(N), we can be sure of
            picking up every class modulo GammaH(N) since this
            contains Gamma1(N); and the reduce_cusp call does the
            checking to make sure we don't get any duplicates.

        EXAMPLES::

            sage: Gamma1(5)._find_cusps()
            [0, 2/5, 1/2, Infinity]
            sage: Gamma1(35)._find_cusps()
            [0, 2/35, 1/17, 1/16, 1/15, 1/14, 1/13, 1/12, 3/35, 1/11, 1/10, 1/9, 4/35, 1/8, 2/15, 1/7, 1/6, 6/35, 1/5, 3/14, 8/35, 1/4, 9/35, 4/15, 2/7, 3/10, 11/35, 1/3, 12/35, 5/14, 13/35, 2/5, 3/7, 16/35, 17/35, 1/2, 8/15, 4/7, 3/5, 9/14, 7/10, 5/7, 11/14, 4/5, 6/7, 9/10, 13/14, Infinity]
            sage: Gamma1(24)._find_cusps() == Gamma1(24).cusps(algorithm='modsym')
            True
            sage: GammaH(24, [13,17])._find_cusps() == GammaH(24,[13,17]).cusps(algorithm='modsym')
            True
        """

        s = []
        hashes = []
        N = self.level()
        
        for d in xrange(1, 1+N):
            w = N.gcd(d)
            M = int(w) if w > 1 else 2
            for a in xrange(1,M):
                if gcd(a, w) != 1: 
                    continue
                while gcd(a, d) != 1:
                    a += w
                c = self.reduce_cusp(Cusp(a,d))
                h = hash(c)
                if not h in hashes:
                    hashes.append(h)
                    s.append(c)
        return sorted(s)
       
    def __call__(self, x, check=True):
        r"""
        Create an element of this congruence subgroup from x.

        If the optional flag check is True (default), check whether
        x actually gives an element of self.

        EXAMPLES::

            sage: G = GammaH(10, [3])
            sage: G([1, 0, -10, 1])
            [ 1   0]
            [-10  1]
            sage: G(matrix(ZZ, 2, [7, 1, 20, 3]))
            [ 7  1]
            [20  3]
            sage: GammaH(10, [9])([7, 1, 20, 3])
            Traceback (most recent call last):
            ...
            TypeError: matrix must have lower right entry (=3) congruent modulo 10 to some element of H
        """
        from all import SL2Z
        x = SL2Z(x, check)
        if not check:
            return x
        
        c = x.c()
        d = x.d()
        N = self.level()
        if c%N != 0:
            raise TypeError, "matrix must have lower left entry (=%s) divisible by %s" %(c, N)
        elif d%N in self._list_of_elements_in_H():
            return x
        else:
            raise TypeError, "matrix must have lower right entry (=%s) congruent modulo %s to some element of H" %(d, N)

    def gamma0_coset_reps(self):
        r"""
        Return a set of coset representatives for self \\ Gamma0(N), where N is
        the level of self.

        EXAMPLE::

            sage: GammaH(108, [1,-1]).gamma0_coset_reps()
            [
            [1 0]  [-43 -45]  [ 31  33]  [-49 -54]  [ 25  28]  [-19 -22]
            [0 1], [108 113], [108 115], [108 119], [108 121], [108 125],
            <BLANKLINE>
            [-17 -20]  [ 47  57]  [ 13  16]  [ 41  52]  [  7   9]  [-37 -49]
            [108 127], [108 131], [108 133], [108 137], [108 139], [108 143],
            <BLANKLINE>
            [-35 -47]  [ 29  40]  [ -5  -7]  [ 23  33]  [-11 -16]  [ 53  79]
            [108 145], [108 149], [108 151], [108 155], [108 157], [108 161]
            ]
        """
        from all import Gamma0
        N = self.level()
        G = Gamma0(N)
        s = []
        return [G(lift_to_sl2z(0, d.lift(), N)) for d in _GammaH_coset_helper(N, self._list_of_elements_in_H())]

    def coset_reps(self):
        r"""
        Return a set of coset representatives for self \\ SL2Z.

        EXAMPLES::

            sage: list(Gamma1(3).coset_reps())
            [
            [1 0]  [-1 -2]  [ 0 -1]  [-2  1]  [1 0]  [-3 -2]  [ 0 -1]  [-2 -3]
            [0 1], [ 3  5], [ 1  0], [ 5 -3], [1 1], [ 8  5], [ 1  2], [ 5  7]
            ]
            sage: len(list(Gamma1(31).coset_reps())) == 31**2 - 1
            True
        """
        from all import Gamma0, SL2Z
        reps1 = Gamma0(self.level()).coset_reps()
        for r in reps1:
            reps2 = self.gamma0_coset_reps()
            for t in reps2:
                yield SL2Z(t)*r


    def is_subgroup(self, other):
        r"""
        Return True if self is a subgroup of right, and False
        otherwise.

        EXAMPLES::

            sage: GammaH(24,[7]).is_subgroup(SL2Z)
            True
            sage: GammaH(24,[7]).is_subgroup(Gamma0(8))
            True
            sage: GammaH(24, []).is_subgroup(GammaH(24, [7]))
            True
            sage: GammaH(24, []).is_subgroup(Gamma1(24))
            True
            sage: GammaH(24, [17]).is_subgroup(GammaH(24, [7]))
            False
            sage: GammaH(1371, [169]).is_subgroup(GammaH(457, [169]))
            True
        """

        from all import is_Gamma0, is_Gamma1
        if not isinstance(other, GammaH_class):
            raise NotImplementedError

        # level of self should divide level of other
        if self.level() % other.level():
            return False

        # easy cases
        if is_Gamma0(other):
            return True # recall self is a GammaH, so it's contained in Gamma0

        if is_Gamma1(other) and len(self._generators_for_H()) > 0:
            return False

        else:
            # difficult case
            t = other._list_of_elements_in_H()
            for x in self._generators_for_H():
                if not (x in t):
                    return False
            return True


    def index(self):
        r"""
        Return the index of self in SL2Z.

        EXAMPLE::

            sage: [G.index() for G in Gamma0(40).gamma_h_subgroups()]
            [72, 144, 144, 144, 144, 288, 288, 288, 288, 144, 288, 288, 576, 576, 144, 288, 288, 576, 576, 144, 288, 288, 576, 576, 288, 576, 1152]
        """
        from all import Gamma1
        return Gamma1(self.level()).index() / len(self._list_of_elements_in_H())

    def nu2(self):
        r"""
        Return the number of orbits of elliptic points of order 2 for this
        group.

        EXAMPLE::

            sage: [H.nu2() for n in [1..10] for H in Gamma0(n).gamma_h_subgroups()]
            [1, 1, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0]
            sage: GammaH(33,[2]).nu2()
            0
            sage: GammaH(5,[2]).nu2()
            2        
    
        AUTHORS:

        - Jordi Quer

        """
        N = self.level()
        H = self._list_of_elements_in_H()
        if N % 4 == 0: return ZZ(0)
        for p, r in N.factor():
            if p % 4 == 3: return ZZ(0)
        return (euler_phi(N) // len(H))*len([x for x in H if (x**2 + 1) % N == 0])

    def nu3(self):
        r"""
        Return the number of orbits of elliptic points of order 3 for this
        group.

        EXAMPLE::

            sage: [H.nu3() for n in [1..10] for H in Gamma0(n).gamma_h_subgroups()]
            [1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 2, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
            sage: GammaH(33,[2]).nu3()
            0
            sage: GammaH(7,[2]).nu3()
            2        
    
        AUTHORS:

        - Jordi Quer

        """
        N = self.level()
        H = self._list_of_elements_in_H()
        if N % 9 == 0: return ZZ(0)
        for p, r in N.factor():
            if p % 3 == 2: return ZZ(0)
        lenHpm = len(H)
        if N - ZZ(1) not in H: lenHpm*=2
        return (euler_phi(N)//lenHpm)*len([x for x in H if (x**2+x+1) % N == 0])

    def ncusps(self):
        r"""
        Return the number of orbits of cusps (regular or otherwise) for this subgroup.

        EXAMPLE::

            sage: GammaH(33,[2]).ncusps()
            8
            sage: GammaH(32079, [21676]).ncusps()
            28800

        AUTHORS:
        
        - Jordi Quer

        """
        N = self.level()
        H = self._list_of_elements_in_H()
        c = ZZ(0)
        for d in [d for d in N.divisors() if d**2 <= N]:
            Nd = lcm(d,N//d)
            Hd = set([x % Nd for x in H])
            lenHd = len(Hd)
            if Nd-1 not in Hd: lenHd *= 2
            summand = euler_phi(d)*euler_phi(N//d)//lenHd
            if d**2 == N:
                c = c + summand
            else:
                c = c + 2*summand
        return c

    def nregcusps(self):
        r"""
        Return the number of orbits of regular cusps for this subgroup. A cusp is regular
        if we may find a parabolic element generating the stabiliser of that
        cusp whose eigenvalues are both +1 rather than -1. If G contains -1,
        all cusps are regular.

        EXAMPLES::
            
            sage: GammaH(20, [17]).nregcusps()
            4
            sage: GammaH(20, [17]).nirregcusps()
            2
            sage: GammaH(3212, [2045, 2773]).nregcusps()
            1440
            sage: GammaH(3212, [2045, 2773]).nirregcusps()
            720

        AUTHOR:

        - Jordi Quer
        """
        if self.is_even():
            return self.ncusps()

        N = self.level()
        H = self._list_of_elements_in_H()

        c = ZZ(0)
        for d in [d for d in divisors(N) if d**2 <= N]:
            Nd = lcm(d,N//d)
            Hd = set([x%Nd for x in H])
            if Nd - 1 not in Hd:
                summand = euler_phi(d)*euler_phi(N//d)//(2*len(Hd))
                if d**2==N:
                    c = c + summand
                else:
                    c = c + 2*summand
        return c

    def nirregcusps(self):
        r"""
        Return the number of irregular cusps for this subgroup.

        EXAMPLES::

            sage: GammaH(3212, [2045, 2773]).nirregcusps()
            720
        """

        return self.ncusps() - self.nregcusps()

    def dimension_new_cusp_forms(self, k=2, p=0):
        r"""
        Return the dimension of the space of new (or `p`-new)
        weight `k` cusp forms for this congruence subgroup.
    
        INPUT:
    
        -  ``k`` - an integer (default: 2), the weight. Not fully implemented for k = 1.
        -  ``p`` - integer (default: 0); if nonzero, compute the `p`-new subspace.
    
        OUTPUT: Integer
    
        EXAMPLES::
        
            sage: GammaH(33,[2]).dimension_new_cusp_forms()
            3
            sage: Gamma1(4*25).dimension_new_cusp_forms(2, p=5)
            225
            sage: Gamma1(33).dimension_new_cusp_forms(2)
            19
            sage: Gamma1(33).dimension_new_cusp_forms(2,p=11)
            21

        """
        N = self.level()
        if p==0 or N % p != 0:
            return sum([H.dimension_cusp_forms(k) * mumu(N // H.level()) \
                    for H in self.divisor_subgroups()])
        else:
            return self.dimension_cusp_forms(k) - \
                   2*self.restrict(N//p).dimension_new_cusp_forms(k)
    
    def image_mod_n(self):
        r"""
        Return the image of this group in `SL(2, \ZZ / N\ZZ)`.

        EXAMPLE::

            sage: Gamma0(3).image_mod_n()
            Matrix group over Ring of integers modulo 3 with 2 generators: 
             [[[2, 0], [0, 2]], [[1, 1], [0, 1]]]

        TEST::

            sage: for n in [2..20]:
            ...     for g in Gamma0(n).gamma_h_subgroups():
            ...       G = g.image_mod_n()
            ...       assert G.order() == Gamma(n).index() / g.index()
        """
        N = self.level()
        if N == 1: 
            raise NotImplementedError, "Matrix groups over ring of integers modulo 1 not implemented"
        gens = [matrix(Zmod(N), 2, 2, [x, 0, 0, Zmod(N)(1)/x]) for x in self._generators_for_H()]
        gens += [matrix(Zmod(N),2,[1,1,0,1])]
        return MatrixGroup(gens)
        
def _list_subgroup(N, gens):   
    r"""
    Given an integer ``N`` and a list of integers ``gens``, return a list of
    the elements of the subgroup of `(\ZZ / N\ZZ)^\times` generated by the
    elements of ``gens``.

    EXAMPLE::

        sage: sage.modular.arithgroup.congroup_gammaH._list_subgroup(11, [3])
        [1, 3, 4, 5, 9]
    """
    H = set([1])
    N = int(N)
    for g in gens:
        if gcd(g, N) != 1:
            raise ValueError, "gen (=%s) is not in (Z/%sZ)^*"%(g,N)
        gk = int(g) % N
        sbgrp = [gk]
        while not (gk in H):
            gk = (gk * g)%N
            sbgrp.append(gk)
        H = set([(x*h)%N for x in sbgrp for h in H])
    H = list(H)
    H.sort()
    return H

def _GammaH_coset_helper(N, H):
    r""" 
    Return a list of coset representatives for H in (Z / NZ)^*.

    EXAMPLE::

        sage: from sage.modular.arithgroup.congroup_gammaH import _GammaH_coset_helper
        sage: _GammaH_coset_helper(108, [1, 107])
        [1, 5, 7, 11, 13, 17, 19, 23, 25, 29, 31, 35, 37, 41, 43, 47, 49, 53]
    """
    t = [Zmod(N)(1)]
    W = [Zmod(N)(h) for h in H]
    HH = [Zmod(N)(h) for h in H]
    k = euler_phi(N)
    
    for i in xrange(1, N):
        if gcd(i, N) != 1: continue
        if not i in W:
            t.append(t[0]*i)
            W = W + [i*h for h in HH]
            if len(W) == k: break
    return t

def mumu(N):
    """
    Return 0 if any cube divides `N`. Otherwise return
    `(-2)^v` where `v` is the number of primes that
    exactly divide `N`.
    
    This is similar to the Moebius function.
    
    INPUT:
    
    
    -  ``N`` - an integer at least 1
    
    
    OUTPUT: Integer
    
    EXAMPLES::
    
        sage: from sage.modular.arithgroup.congroup_gammaH import mumu
        sage: mumu(27)
        0
        sage: mumu(6*25)
        4
        sage: mumu(7*9*25)
        -2
        sage: mumu(9*25)
        1
    """
    if N < 1:
        raise ValueError, "N must be at least 1"
    p = 1
    for _,r in factor(N):
        if r > 2:
            return ZZ(0)
        elif r == 1:
            p *= -2
    return ZZ(p)