Python all.is_Gamma1函数代码示例

    def newform_decomposition(self, names=None):
        """
        Return the newforms of the simple subvarieties in the decomposition of
        self as a product of simple subvarieties, up to isogeny.

        OUTPUT:

        - an array of newforms

        EXAMPLES::

            sage: J0(81).newform_decomposition('a')
            [q - 2*q^4 + O(q^6), q - 2*q^4 + O(q^6), q + a0*q^2 + q^4 - a0*q^5 + O(q^6)]

            sage: J1(19).newform_decomposition('a')
            [q - 2*q^3 - 2*q^4 + 3*q^5 + O(q^6),
             q + a1*q^2 + (-1/9*a1^5 - 1/3*a1^4 - 1/3*a1^3 + 1/3*a1^2 - a1 - 1)*q^3 + (4/9*a1^5 + 2*a1^4 + 14/3*a1^3 + 17/3*a1^2 + 6*a1 + 2)*q^4 + (-2/3*a1^5 - 11/3*a1^4 - 10*a1^3 - 14*a1^2 - 15*a1 - 9)*q^5 + O(q^6)]
        """
        if self.dimension() == 0:
            return []
        G = self.group()
        if not (is_Gamma0(G) or is_Gamma1(G)):
            return [S.newform(names=names) for S in self.decomposition()]
        Gtype = G.parent()
        N = G.level()
        preans = [Newforms(Gtype(d), names=names) *
                  len(Integer(N/d).divisors()) for d in N.divisors()]
        ans = [newform for l in preans for newform in l]
        return ans

    def eisenstein_params(self):
        """
        Return parameters that define all Eisenstein series in self.

        OUTPUT: an immutable Sequence

        EXAMPLES::

            sage: m = ModularForms(Gamma0(22), 2)
            sage: v = m.eisenstein_params(); v
            [(Dirichlet character modulo 22 of conductor 1 mapping 13 |--> 1, Dirichlet character modulo 22 of conductor 1 mapping 13 |--> 1, 2), (Dirichlet character modulo 22 of conductor 1 mapping 13 |--> 1, Dirichlet character modulo 22 of conductor 1 mapping 13 |--> 1, 11), (Dirichlet character modulo 22 of conductor 1 mapping 13 |--> 1, Dirichlet character modulo 22 of conductor 1 mapping 13 |--> 1, 22)]
            sage: type(v)
            <class 'sage.structure.sequence.Sequence_generic'>
        """
        try:
            return self.__eisenstein_params
        except AttributeError:
            eps = self.character()
            if eps is None:
                if arithgroup.is_Gamma1(self.group()):
                    eps = self.level()
                else:
                    raise NotImplementedError
            params = eis_series.compute_eisenstein_params(eps, self.weight())
            self.__eisenstein_params = Sequence(params, immutable=True)
        return self.__eisenstein_params

    def _dim_eisenstein(self):
        """
        Return the dimension of the Eisenstein subspace of this modular
        symbols space, computed using a dimension formula.

        EXAMPLES::

            sage: m = ModularForms(GammaH(13,[4]), 2); m
            Modular Forms space of dimension 3 for Congruence Subgroup Gamma_H(13) with H generated by [4] of weight 2 over Rational Field
            sage: m._dim_eisenstein()
            3

            sage: m = ModularForms(DirichletGroup(13).0,7); m
            Modular Forms space of dimension 8, character [zeta12] and weight 7 over Cyclotomic Field of order 12 and degree 4
            sage: m._dim_eisenstein()
            2
            sage: m._dim_cuspidal()
            6

        Test that :trac:`24030` is fixed::

            sage: ModularForms(GammaH(40, [21]), 1).dimension() # indirect doctest
            16
        """
        if arithgroup.is_Gamma1(self.group()) and self.character() is not None:
            return self.group().dimension_eis(self.weight(), self.character())
        else:
            return self.group().dimension_eis(self.weight())

    def _p_stabilize_parent_space(self, p, new_base_ring):
        r"""
        Return the space of Pollack-Stevens modular symbols of level
        `p N`, with changed base ring.  This is used internally when
        constructing the `p`-stabilization of a modular symbol.

        INPUT:

        - ``p`` -- prime number
        - ``new_base_ring`` -- the base ring of the result

        OUTPUT:

        The space of modular symbols of level `p N`, where `N` is the level
        of this space.

        EXAMPLES::

            sage: D = OverconvergentDistributions(2, 7)
            sage: M = PollackStevensModularSymbols(Gamma(13), coefficients=D)
            sage: M._p_stabilize_parent_space(7, M.base_ring())
            Space of overconvergent modular symbols for Congruence Subgroup
            Gamma(91) with sign 0 and values in Space of 7-adic distributions
            with k=2 action and precision cap 20

            sage: D = OverconvergentDistributions(4, 17)
            sage: M = PollackStevensModularSymbols(Gamma1(3), coefficients=D)
            sage: M._p_stabilize_parent_space(17, Qp(17))
            Space of overconvergent modular symbols for Congruence
            Subgroup Gamma1(51) with sign 0 and values in Space of
            17-adic distributions with k=4 action and precision cap 20
        """
        N = self.level()
        if N % p == 0:
            raise ValueError("the level is not prime to p")
        from sage.modular.arithgroup.all import (Gamma, is_Gamma, Gamma0,
                                                 is_Gamma0, Gamma1, is_Gamma1)
        G = self.group()
        if is_Gamma0(G):
            G = Gamma0(N * p)
        elif is_Gamma1(G):
            G = Gamma1(N * p)
        elif is_Gamma(G):
            G = Gamma(N * p)
        else:
            raise NotImplementedError
        return PollackStevensModularSymbols(G, coefficients=self.coefficient_module().change_ring(new_base_ring), sign=self.sign())

    def _dim_new_cuspidal(self):
        """
        Return the dimension of the new cuspidal subspace, computed using
        dimension formulas.

        EXAMPLES::

            sage: m = ModularForms(GammaH(11,[2]), 2); m._dim_new_cuspidal()
            1
        """
        try:
            return self.__the_dim_new_cuspidal
        except AttributeError:
            if arithgroup.is_Gamma1(self.group()) and self.character() is not None:
                self.__the_dim_new_cuspidal = self.group().dimension_new_cusp_forms(self.weight(), self.character())
            else:
                self.__the_dim_new_cuspidal = self.group().dimension_new_cusp_forms(self.weight())
        return self.__the_dim_new_cuspidal

    def _dim_cuspidal(self):
        """
        Return the dimension of the cuspidal subspace of this ambient
        modular forms space, computed using a dimension formula.

        EXAMPLES::

            sage: m = ModularForms(GammaH(11,[4]), 2); m
            Modular Forms space of dimension 2 for Congruence Subgroup Gamma_H(11) with H generated by [4] of weight 2 over Rational Field
            sage: m._dim_cuspidal()
            1
        """
        try:
            return self.__the_dim_cuspidal
        except AttributeError:
            if arithgroup.is_Gamma1(self.group()) and self.character() is not None:
               self.__the_dim_cuspidal = self.group().dimension_cusp_forms(self.weight(), self.character())
            else:
               self.__the_dim_cuspidal = self.group().dimension_cusp_forms(self.weight())
        return self.__the_dim_cuspidal

    def _dim_new_cuspidal(self):
        """
        Return the dimension of the new cuspidal subspace, computed using
        dimension formulas.

        EXAMPLES::

            sage: m = ModularForms(GammaH(11,[2]), 2); m._dim_new_cuspidal()
            1
            sage: m = ModularForms(DirichletGroup(33).0,7); m
            Modular Forms space of dimension 26, character [-1, 1] and weight 7 over Rational Field
            sage: m._dim_new_cuspidal()
            20
            sage: m._dim_cuspidal()
            22
        """
        if arithgroup.is_Gamma1(self.group()) and self.character() is not None:
            return self.group().dimension_new_cusp_forms(self.weight(), self.character())
        else:
            return self.group().dimension_new_cusp_forms(self.weight())

    def label(self):
        """
        Return canonical label that defines this newform modular
        abelian variety.

        OUTPUT:
            string

        EXAMPLES::

            sage: A = AbelianVariety('43b')
            sage: A.label()
            '43b'
        """
        G = self.__f.group()
        if is_Gamma0(G):
            group = ''
        elif is_Gamma1(G):
            group = 'G1'
        elif is_GammaH(G):
            group = 'GH[' + ','.join([str(z) for z in G._generators_for_H()]) + ']'
        return '%s%s%s'%(self.level(), cremona_letter_code(self.factor_number()), group)

    def _dim_eisenstein(self):
        """
        Return the dimension of the Eisenstein subspace of this modular
        symbols space, computed using a dimension formula.

        EXAMPLES::

            sage: m = ModularForms(GammaH(13,[4]), 2); m
            Modular Forms space of dimension 3 for Congruence Subgroup Gamma_H(13) with H generated by [4] of weight 2 over Rational Field
            sage: m._dim_eisenstein()
            3
        """
        try:
            return self.__the_dim_eisenstein
        except AttributeError:
            if self.weight() == 1:
                self.__the_dim_eisenstein = len(self.eisenstein_params())
            else:
                if arithgroup.is_Gamma1(self.group()) and self.character() is not None:
                    self.__the_dim_eisenstein = self.group().dimension_eis(self.weight(), self.character())
                else:
                    self.__the_dim_eisenstein = self.group().dimension_eis(self.weight())
        return self.__the_dim_eisenstein

    def _dim_cuspidal(self):
        """
        Return the dimension of the cuspidal subspace of this ambient
        modular forms space, computed using a dimension formula.

        EXAMPLES::

            sage: m = ModularForms(GammaH(11,[3]), 2); m
            Modular Forms space of dimension 2 for Congruence Subgroup Gamma_H(11) with H generated by [3] of weight 2 over Rational Field
            sage: m._dim_cuspidal()
            1
            sage: m = ModularForms(DirichletGroup(389,CyclotomicField(4)).0,3); m._dim_cuspidal()
            64
        """
        if self._eis_only:
            return 0
        if arithgroup.is_Gamma1(self.group()) and self.character() is not None:
            if self.weight() != 1:
                return self.group().dimension_cusp_forms(self.weight(), self.character())
            else:
                return weight1.dimension_cusp_forms(self.character())
        else:
           return self.group().dimension_cusp_forms(self.weight())

    def multiple_of_order_using_frobp(self, maxp=None):
        """
        Return a multiple of the order of this torsion group.

        In the `Gamma_0` case, the multiple is computed using characteristic
        polynomials of Hecke operators of odd index not dividing the level. In
        the `Gamma_1` case, the multiple is computed by expressing the
        frobenius polynomial in terms of the characteristic polynomial of left
        multiplication by `a_p` for odd primes p not dividing the level.

        INPUT:


        -  ``maxp`` - (default: None) If maxp is None (the
           default), return gcd of best bound computed so far with bound
           obtained by computing GCD's of orders modulo p until this gcd
           stabilizes for 3 successive primes. If maxp is given, just use all
           primes up to and including maxp.


        EXAMPLES::

            sage: J = J0(11)
            sage: G = J.rational_torsion_subgroup()
            sage: G.multiple_of_order_using_frobp(11)
            5

        Increasing maxp may yield a tighter bound. If maxp=None, then Sage
        will use more primes until the multiple stabilizes for 3 successive
        primes.  ::

            sage: J = J0(389)
            sage: G = J.rational_torsion_subgroup(); G
            Torsion subgroup of Abelian variety J0(389) of dimension 32
            sage: G.multiple_of_order_using_frobp()
            97
            sage: [G.multiple_of_order_using_frobp(p) for p in prime_range(3,11)]
            [92645296242160800, 7275, 291]
            sage: [G.multiple_of_order_using_frobp(p) for p in prime_range(3,13)]
            [92645296242160800, 7275, 291, 97]
            sage: [G.multiple_of_order_using_frobp(p) for p in prime_range(3,19)]
            [92645296242160800, 7275, 291, 97, 97, 97]

        We can compute the multiple of order of the torsion subgroup for Gamma0
        and Gamma1 varieties, and their products. ::

            sage: A = J0(11) * J0(33)
            sage: A.rational_torsion_subgroup().multiple_of_order_using_frobp()
            1000

            sage: A = J1(23)
            sage: A.rational_torsion_subgroup().multiple_of_order_using_frobp()
            9406793
            sage: A.rational_torsion_subgroup().multiple_of_order_using_frobp(maxp=50)
            408991

            sage: A = J1(19) * J0(21)
            sage: A.rational_torsion_subgroup().multiple_of_order_using_frobp()
            35064

        The next example illustrates calling this function with a larger
        input and how the result may be cached when maxp is None::

            sage: T = J0(43)[1].rational_torsion_subgroup()
            sage: T.multiple_of_order_using_frobp()
            14
            sage: T.multiple_of_order_using_frobp(50)
            7
            sage: T.multiple_of_order_using_frobp()
            7

        This function is not implemented for general congruence subgroups
        unless the dimension is zero. ::

            sage: A = JH(13,[2]); A
            Abelian variety J0(13) of dimension 0
            sage: A.rational_torsion_subgroup().multiple_of_order_using_frobp()
            1

            sage: A = JH(15, [2]); A
            Abelian variety JH(15,[2]) of dimension 1
            sage: A.rational_torsion_subgroup().multiple_of_order_using_frobp()
            Traceback (most recent call last):
            ...
            NotImplementedError: torsion multiple only implemented for Gamma0 and Gamma1
        """
        if maxp is None:
            try:
                return self.__multiple_of_order_using_frobp
            except AttributeError:
                pass
        A = self.abelian_variety()
        if A.dimension() == 0:
            T = ZZ(1)
            self.__multiple_of_order_using_frobp = T
            return T
        if not all((is_Gamma0(G) or is_Gamma1(G) for G in A.groups())):
            raise NotImplementedError("torsion multiple only implemented for Gamma0 and Gamma1")

        bnd = ZZ(0)
#.........这里部分代码省略.........

    def multiple_of_order(self, maxp=None, proof=True):
        """
        Return a multiple of the order.

        INPUT:

        - ``proof`` -- a boolean (default: True)

        The computation of the rational torsion order of J1(p) is conjectural
        and will only be used if proof=False. See Section 6.2.3 of [CES2003]_.

        EXAMPLES::

            sage: J = J1(11); J
            Abelian variety J1(11) of dimension 1
            sage: J.rational_torsion_subgroup().multiple_of_order()
            5

            sage: J = J0(17)
            sage: J.rational_torsion_subgroup().order()
            4

        This is an example where proof=False leads to a better bound and better
        performance. ::

            sage: J = J1(23)
            sage: J.rational_torsion_subgroup().multiple_of_order() # long time (2s)
            9406793
            sage: J.rational_torsion_subgroup().multiple_of_order(proof=False)
            408991
        """

        try:
            if proof:
                return self._multiple_of_order
            else:
                return self._multiple_of_order_proof_false
        except AttributeError:
            pass

        A = self.abelian_variety()
        N = A.level()

        if A.dimension() == 0:
            self._multiple_of_order = ZZ(1)
            self._multiple_of_order_proof_false = self._multiple_of_order
            return self._multiple_of_order

        # return the order of the cuspidal subgroup in the J0(p) case
        if A.is_J0() and N.is_prime():
            self._multiple_of_order = QQ((A.level()-1)/12).numerator()
            self._multiple_of_order_proof_false = self._multiple_of_order
            return self._multiple_of_order

        # The elliptic curve case
        if A.dimension() == 1:
            self._multiple_of_order = A.elliptic_curve().torsion_order()
            self._multiple_of_order_proof_false = self._multiple_of_order
            return self._multiple_of_order

        # The conjectural J1(p) case
        if not proof and A.is_J1() and N.is_prime():
            epsilons = [epsilon for epsilon in DirichletGroup(N)
                        if not epsilon.is_trivial() and epsilon.is_even()]
            bernoullis = [epsilon.bernoulli(2) for epsilon in epsilons]
            self._multiple_of_order_proof_false = ZZ(N/(2**(N-3))*prod(bernoullis))
            return self._multiple_of_order_proof_false

        # The Gamma0 and Gamma1 case
        if all((is_Gamma0(G) or is_Gamma1(G) for G in A.groups())):
            self._multiple_of_order = self.multiple_of_order_using_frobp()
            return self._multiple_of_order

        # Unhandled case
        raise NotImplementedError("No implemented algorithm")

def canonical_parameters(group, level, weight, base_ring):
    """
    Given a group, level, weight, and base_ring as input by the user,
    return a canonicalized version of them, where level is a Sage
    integer, group really is a group, weight is a Sage integer, and
    base_ring a Sage ring. Note that we can't just get the level from
    the group, because we have the convention that the character for
    Gamma1(N) is None (which makes good sense).

    INPUT:


    -  ``group`` - int, long, Sage integer, group,
       dirichlet character, or

    -  ``level`` - int, long, Sage integer, or group

    -  ``weight`` - coercible to Sage integer

    -  ``base_ring`` - commutative Sage ring


    OUTPUT:


    -  ``level`` - Sage integer

    -  ``group`` - congruence subgroup

    -  ``weight`` - Sage integer

    -  ``ring`` - commutative Sage ring


    EXAMPLES::

        sage: from sage.modular.modform.constructor import canonical_parameters
        sage: v = canonical_parameters(5, 5, int(7), ZZ); v
        (5, Congruence Subgroup Gamma0(5), 7, Integer Ring)
        sage: type(v[0]), type(v[1]), type(v[2]), type(v[3])
        (<type 'sage.rings.integer.Integer'>,
         <class 'sage.modular.arithgroup.congroup_gamma0.Gamma0_class_with_category'>,
         <type 'sage.rings.integer.Integer'>,
         <type 'sage.rings.integer_ring.IntegerRing_class'>)
        sage: canonical_parameters( 5, 7, 7, ZZ )
        Traceback (most recent call last):
        ...
        ValueError: group and level do not match.
    """
    weight = rings.Integer(weight)
    if weight <= 0:
        raise NotImplementedError, "weight must be at least 1"

    if isinstance(group, dirichlet.DirichletCharacter):
        if ( group.level() != rings.Integer(level) ):
            raise ValueError, "group.level() and level do not match."
        group = group.minimize_base_ring()
        level = rings.Integer(level)

    elif arithgroup.is_CongruenceSubgroup(group):
        if ( rings.Integer(level) != group.level() ):
            raise ValueError, "group.level() and level do not match."
        # normalize the case of SL2Z
        if arithgroup.is_SL2Z(group) or \
           arithgroup.is_Gamma1(group) and group.level() == rings.Integer(1):
            group = arithgroup.Gamma0(rings.Integer(1))

    elif group is None:
        pass

    else:
        try:
            m = rings.Integer(group)
        except TypeError:
            raise TypeError, "group of unknown type."
        level = rings.Integer(level)
        if ( m != level ):
            raise ValueError, "group and level do not match."
        group = arithgroup.Gamma0(m)

    if not is_CommutativeRing(base_ring):
        raise TypeError, "base_ring (=%s) must be a commutative ring"%base_ring

    # it is *very* important to include the level as part of the data
    # that defines the key, since dirichlet characters of different
    # levels can compare equal, but define much different modular
    # forms spaces.
    return level, group, weight, base_ring

#.........这里部分代码省略.........

    We create some weight 1 spaces. The first example works fine, since we can prove purely by Riemann surface theory that there are no weight 1 cusp forms::

        sage: M = ModularForms(Gamma1(11), 1); M
        Modular Forms space of dimension 5 for Congruence Subgroup Gamma1(11) of weight 1 over Rational Field
        sage: M.basis()
        [
        1 + 22*q^5 + O(q^6),
        q + 4*q^5 + O(q^6),
        q^2 - 4*q^5 + O(q^6),
        q^3 - 5*q^5 + O(q^6),
        q^4 - 3*q^5 + O(q^6)
        ]
        sage: M.cuspidal_subspace().basis()
        [
        ]
        sage: M == M.eisenstein_subspace()
        True

    This example doesn't work so well, because we can't calculate the cusp
    forms; but we can still work with the Eisenstein series.

        sage: M = ModularForms(Gamma1(57), 1); M
        Modular Forms space of dimension (unknown) for Congruence Subgroup Gamma1(57) of weight 1 over Rational Field
        sage: M.basis()
        Traceback (most recent call last):
        ...
        NotImplementedError: Computation of dimensions of weight 1 cusp forms spaces not implemented in general
        sage: M.cuspidal_subspace().basis()
        Traceback (most recent call last):
        ...
        NotImplementedError: Computation of dimensions of weight 1 cusp forms spaces not implemented in general

        sage: E = M.eisenstein_subspace(); E
        Eisenstein subspace of dimension 36 of Modular Forms space of dimension (unknown) for Congruence Subgroup Gamma1(57) of weight 1 over Rational Field
        sage: (E.0 + E.2).q_expansion(40)
        1 + q^2 + 1473/2*q^36 - 1101/2*q^37 + q^38 - 373/2*q^39 + O(q^40)

    """
    if isinstance(group, dirichlet.DirichletCharacter):
        if base_ring is None:
            base_ring = group.minimize_base_ring().base_ring()
    if base_ring is None:
        base_ring = rings.QQ

    if isinstance(group, dirichlet.DirichletCharacter) \
           or arithgroup.is_CongruenceSubgroup(group):
        level = group.level()
    else:
        level = group

    key = canonical_parameters(group, level, weight, base_ring)

    if use_cache and _cache.has_key(key):
         M = _cache[key]()
         if not (M is None):
             M.set_precision(prec)
             return M

    (level, group, weight, base_ring) = key

    M = None
    if arithgroup.is_Gamma0(group):
        M = ambient_g0.ModularFormsAmbient_g0_Q(group.level(), weight)
        if base_ring != rings.QQ:
            M = ambient_R.ModularFormsAmbient_R(M, base_ring)

    elif arithgroup.is_Gamma1(group):
        M = ambient_g1.ModularFormsAmbient_g1_Q(group.level(), weight)
        if base_ring != rings.QQ:
            M = ambient_R.ModularFormsAmbient_R(M, base_ring)

    elif arithgroup.is_GammaH(group):
        M = ambient_g1.ModularFormsAmbient_gH_Q(group, weight)
        if base_ring != rings.QQ:
            M = ambient_R.ModularFormsAmbient_R(M, base_ring)

    elif isinstance(group, dirichlet.DirichletCharacter):
        eps = group
        if eps.base_ring().characteristic() != 0:
            # TODO -- implement this
            # Need to add a lift_to_char_0 function for characters,
            # and need to still remember eps.
            raise NotImplementedError, "currently the character must be over a ring of characteristic 0."
        eps = eps.minimize_base_ring()
        if eps.is_trivial():
            return ModularForms(eps.modulus(), weight, base_ring,
                                use_cache = use_cache,
                                prec = prec)
        M = ambient_eps.ModularFormsAmbient_eps(eps, weight)
        if base_ring != eps.base_ring():
            M = M.base_extend(base_ring) # ambient_R.ModularFormsAmbient_R(M, base_ring)

    if M is None:
        raise NotImplementedError, \
           "computation of requested space of modular forms not defined or implemented"

    M.set_precision(prec)
    _cache[key] = weakref.ref(M)
    return M

#.........这里部分代码省略.........
    We create a space of modular symbols with nontrivial character in
    characteristic 2.

    ::

        sage: G = DirichletGroup(13,GF(4,'a')); G
        Group of Dirichlet characters modulo 13 with values in Finite Field in a of size 2^2
        sage: e = G.list()[2]; e
        Dirichlet character modulo 13 of conductor 13 mapping 2 |--> a + 1
        sage: M = ModularSymbols(e,4); M
        Modular Symbols space of dimension 8 and level 13, weight 4, character [a + 1], sign 0, over Finite Field in a of size 2^2
        sage: M.basis()
        ([X*Y,(1,0)], [X*Y,(1,5)], [X*Y,(1,10)], [X*Y,(1,11)], [X^2,(0,1)], [X^2,(1,10)], [X^2,(1,11)], [X^2,(1,12)])
        sage: M.T(2).matrix()
        [    0     0     0     0     0     0     1     1]
        [    0     0     0     0     0     0     0     0]
        [    0     0     0     0     0 a + 1     1     a]
        [    0     0     0     0     0     1 a + 1     a]
        [    0     0     0     0 a + 1     0     1     1]
        [    0     0     0     0     0     a     1     a]
        [    0     0     0     0     0     0 a + 1     a]
        [    0     0     0     0     0     0     1     0]

    We illustrate the custom_init function, which can be used to make
    arbitrary changes to the modular symbols object before its
    presentation is computed::

        sage: ModularSymbols_clear_cache()
        sage: def custom_init(M):
        ....:     M.customize='hi'
        sage: M = ModularSymbols(1,12, custom_init=custom_init)
        sage: M.customize
        'hi'

    We illustrate the relation between custom_init and use_cache::

        sage: def custom_init(M):
        ....:     M.customize='hi2'
        sage: M = ModularSymbols(1,12, custom_init=custom_init)
        sage: M.customize
        'hi'
        sage: M = ModularSymbols(1,12, custom_init=custom_init, use_cache=False)
        sage: M.customize
        'hi2'

    TESTS:

    We test use_cache::

        sage: ModularSymbols_clear_cache()
        sage: M = ModularSymbols(11,use_cache=False)
        sage: sage.modular.modsym.modsym._cache
        {}
        sage: M = ModularSymbols(11,use_cache=True)
        sage: sage.modular.modsym.modsym._cache
        {(Congruence Subgroup Gamma0(11), 2, 0, Rational Field): <weakref at ...; to 'ModularSymbolsAmbient_wt2_g0_with_category' at ...>}
        sage: M is ModularSymbols(11,use_cache=True)
        True
        sage: M is ModularSymbols(11,use_cache=False)
        False
    """
    from . import ambient
    key = canonical_parameters(group, weight, sign, base_ring)

    if use_cache and key in _cache:
         M = _cache[key]()
         if not (M is None): return M

    (group, weight, sign, base_ring) = key

    M = None
    if arithgroup.is_Gamma0(group):
            if weight == 2:
                M = ambient.ModularSymbolsAmbient_wt2_g0(
                    group.level(),sign, base_ring, custom_init=custom_init)
            else:
                M = ambient.ModularSymbolsAmbient_wtk_g0(
                    group.level(), weight, sign, base_ring, custom_init=custom_init)

    elif arithgroup.is_Gamma1(group):

        M = ambient.ModularSymbolsAmbient_wtk_g1(group.level(),
                            weight, sign, base_ring, custom_init=custom_init)

    elif arithgroup.is_GammaH(group):

        M = ambient.ModularSymbolsAmbient_wtk_gamma_h(group,
                            weight, sign, base_ring, custom_init=custom_init)

    elif isinstance(group, tuple):
        eps = group[0]
        M = ambient.ModularSymbolsAmbient_wtk_eps(eps,
                            weight, sign, base_ring, custom_init=custom_init)

    if M is None:
        raise NotImplementedError("computation of requested space of modular symbols not defined or implemented")

    if use_cache:
        _cache[key] = weakref.ref(M)
    return M