def _eval_(self, n, x):
        """
        EXAMPLES::

            sage: y=var('y')
            sage: bessel_I(y,x)
            bessel_I(y, x)
            sage: bessel_I(0.0, 1.0)
            1.26606587775201
            sage: bessel_I(1/2, 1)
            sqrt(2)*sinh(1)/sqrt(pi)
            sage: bessel_I(-1/2, pi)
            sqrt(2)*cosh(pi)/pi
        """
        if (not isinstance(n, Expression) and not isinstance(x, Expression) and
                (is_inexact(n) or is_inexact(x))):
            coercion_model = get_coercion_model()
            n, x = coercion_model.canonical_coercion(n, x)
            return self._evalf_(n, x, parent(n))

        # special identities
        if n == Integer(1) / Integer(2):
            return sqrt(2 / (pi * x)) * sinh(x)
        elif n == -Integer(1) / Integer(2):
            return sqrt(2 / (pi * x)) * cosh(x)

        return None  # leaves the expression unevaluated

    def _eval_(self, n, m, theta, phi, **kwargs):
        r"""
        TESTS::

            sage: x, y = var('x y')
            sage: spherical_harmonic(1, 2, x, y)
            0
            sage: spherical_harmonic(1, -2, x, y)
            0
            sage: spherical_harmonic(1/2, 2, x, y)
            spherical_harmonic(1/2, 2, x, y)
            sage: spherical_harmonic(3, 2, x, y)
            15/4*sqrt(7/30)*cos(x)*e^(2*I*y)*sin(x)^2/sqrt(pi)
            sage: spherical_harmonic(3, 2, 1, 2)
            15/4*sqrt(7/30)*cos(1)*e^(4*I)*sin(1)^2/sqrt(pi)
            sage: spherical_harmonic(3 + I, 2., 1, 2)
            -0.351154337307488 - 0.415562233975369*I
        """
        from sage.structure.coerce import parent

        cc = get_coercion_model().canonical_coercion
        coerced = cc(phi, cc(theta, cc(n, m)[0])[0])[0]
        if is_inexact(coerced) and not isinstance(coerced, Expression):
            return self._evalf_(n, m, theta, phi, parent=parent(coerced))
        elif n in ZZ and m in ZZ and n > -1:
            if abs(m) > n:
                return ZZ(0)
            return meval("spherical_harmonic({},{},{},{})".format(ZZ(n), ZZ(m), maxima(theta), maxima(phi)))
        return

    def __eq__(self, other):
        """
        Check equality.

        TESTS::

            sage: R.<x,y,z> =  QQ[]
            sage: W = DifferentialWeylAlgebra(R)
            sage: dx,dy,dz = W.differentials()
            sage: dy*(x^3-y*z)*dx == -z*dx + x^3*dx*dy - y*z*dx*dy
            True
            sage: W.zero() == 0
            True
            sage: W.one() == 1
            True
            sage: x == 1
            False
            sage: x + 1 == 1
            False
            sage: W(x^3 - y*z) == x^3 - y*z
            True
        """
        if have_same_parent(self, other):
            return self.__monomials == other.__monomials
        try:
            return get_coercion_model().bin_op(self, other, operator.eq)
        except TypeError:
            return False

    def __add__(self, other):
        """
        Return the sum of two subgroups.

        EXAMPLES::

            sage: C = J0(22).cuspidal_subgroup()
            sage: C.gens()
            [[(1/5, 1/5, 4/5, 0)], [(0, 0, 0, 1/5)]]
            sage: A = C.subgroup([C.0]); B = C.subgroup([C.1])
            sage: A + B == C
            True
        """
        if not isinstance(other, FiniteSubgroup):
            raise TypeError("only addition of two finite subgroups is defined")
        A = self.abelian_variety()
        B = other.abelian_variety()
        if not A.in_same_ambient_variety(B):
            raise ValueError("self and other must be in the same ambient Jacobian")
        K = get_coercion_model().common_parent(self.field_of_definition(), other.field_of_definition())
        lattice = self.lattice() + other.lattice()
        if A != B:
            lattice += C.lattice()

        return FiniteSubgroup_lattice(self.abelian_variety(), lattice, field_of_definition=K)

    def tensor_product(self, other):
        r"""
        Return the graded tensor product.

        INPUT:

        - ``other`` -- a filtered vector space.

        OUTPUT:

        The graded tensor product, that is, the tensor product of a
        generator of degree `d_1` with a generator in degree `d_2` has
        degree `d_1 + d_2`.

        EXAMPLES::

            sage: F1 = FilteredVectorSpace(1, 1)
            sage: F2 = FilteredVectorSpace(1, 2)
            sage: F1.tensor_product(F2)
            QQ^1 >= 0
            sage: F1 * F2
            QQ^1 >= 0

            sage: F1.min_degree()
            1
            sage: F2.min_degree()
            2
            sage: (F1*F2).min_degree()
            3

        A suitable base ring is chosen if they do not match::

            sage: v = [(1,0), (0,1)]
            sage: F1 = FilteredVectorSpace(v, {0:[0], 1:[1]}, base_ring=QQ)
            sage: F2 = FilteredVectorSpace(v, {0:[0], 1:[1]}, base_ring=RDF)
            sage: F1 * F2
            RDF^4 >= RDF^3 >= RDF^1 >= 0
        """
        V = self
        W = other
        from sage.structure.element import get_coercion_model
        base_ring = get_coercion_model().common_parent(V.base_ring(), W.base_ring())
        from sage.modules.tensor_operations import VectorCollection, TensorOperation
        V_generators, V_indices = V.presentation()
        W_generators, W_indices = W.presentation()
        V_coll = VectorCollection(V_generators, base_ring, V.dimension())
        W_coll = VectorCollection(W_generators, base_ring, W.dimension())
        T = TensorOperation([V_coll, W_coll], 'product')

        filtration = dict()
        for V_deg in V.support():
            for W_deg in W.support():
                deg = V_deg + W_deg
                indices = filtration.get(deg, set())
                for i in V_indices[V_deg]:
                    for j in W_indices[W_deg]:
                        i_tensor_j = T.index_map(i, j)
                        indices.add(i_tensor_j)
                filtration[deg] = indices
        return FilteredVectorSpace(T.vectors(), filtration, base_ring=base_ring)

        def __mul__(self, right):
            r"""
            Product of two elements

            INPUT:

            - ``self``, ``right`` -- two elements

            This calls the `_mul_` method of ``self``, if it is
            available and the two elements have the same parent.

            Otherwise, the job is delegated to the coercion model.

            Do not override; instead implement a ``_mul_`` method in the
            element class or a ``product`` method in the parent class.

            EXAMPLES::

                sage: S = Semigroups().example("free")
                sage: x = S('a'); y = S('b')
                sage: x * y
                'ab'
            """
            if have_same_parent(self, right) and hasattr(self, "_mul_"):
                return self._mul_(right)
            from sage.structure.element import get_coercion_model
            import operator
            return get_coercion_model().bin_op(self, right, operator.mul)

    def weight_lattice_realization(self):
        r"""
        Return the weight lattice realization used to express weights.

        The weight lattice realization is the common parent which all
        weight lattice realizations of the crystals of ``self`` coerce
        into.

        EXAMPLES::

            sage: LaZ = RootSystem(['A',2,1]).weight_lattice(extended=True).fundamental_weights()
            sage: LaQ = RootSystem(['A',2,1]).weight_space(extended=True).fundamental_weights()
            sage: B = crystals.LSPaths(LaQ[1])
            sage: B.weight_lattice_realization()
            Extended weight space over the Rational Field of the Root system of type ['A', 2, 1]
            sage: C = crystals.AlcovePaths(LaZ[1])
            sage: C.weight_lattice_realization()
            Extended weight lattice of the Root system of type ['A', 2, 1]
            sage: D = crystals.DirectSum([B,C])
            sage: D.weight_lattice_realization()
            Extended weight space over the Rational Field of the Root system of type ['A', 2, 1]
        """
        cm = get_coercion_model()
        return cm.common_parent(*[crystal.weight_lattice_realization()
                                  for crystal in self.crystals])

    def _evalf_(self, n, x, parent=None, algorithm=None):
        """
        EXAMPLES::

            sage: bessel_J(0.0, 1.0)
            0.765197686557967
            sage: bessel_J(0, 1).n(digits=20)
            0.76519768655796655145
            sage: bessel_J(0.5, 1.5)
            0.649838074753747

        Check for correct rounding (:trac:`17122`)::

            sage: R = RealField(113)
            sage: a = R("8.935761195587725798762818805462843676e-01")
            sage: aa = RealField(200)(a)
            sage: for n in [-10..10]:
            ....:     b = bessel_J(R(n), a)
            ....:     bb = R(bessel_J(n, aa))
            ....:     if b != bb:
            ....:         print n, b-bb
        """
        if parent is not None:
            x = parent(x)

        try:
            return x.jn(Integer(n))
        except Exception:
            pass

        n, x = get_coercion_model().canonical_coercion(n, x)
        import mpmath
        return mpmath_utils.call(mpmath.besselj, n, x, parent=parent)

    def __init__(self, surface, m, ring=None):

        if surface.is_mutable():
            if surface.is_finite():
                self._s=surface.copy()
            else:
                raise ValueError("Can not apply matrix to mutable infinite surface.")
        else:
            self._s=surface

        det = m.determinant()

        if det>0:
            self._det_sign=1
        elif det<0:
            self._det_sign=-1
        else:
            raise ValueError("Can not apply matrix with zero determinant to surface.")

        self._m=m

        if ring is None:
            if m.base_ring() == self._s.base_ring():
                base_ring = self._s.base_ring()
            else:
                from sage.structure.element import get_coercion_model
                cm = get_coercion_model()
                base_ring = cm.common_parent(m.base_ring(), self._s.base_ring())
        else:
            base_ring=ring

        self._P=Polygons(base_ring)

        Surface.__init__(self, base_ring, self._s.base_label(), finite=self._s.is_finite())

    def _evalf_try_(self, a, b, z):
        """
        Call :meth:`_evalf_` if one of the arguments is numerical and none
        of the arguments are symbolic.

        OUTPUT:

        - ``None`` if we didn't succeed to call :meth:`_evalf_` or if
          the input wasn't suitable for it.

        - otherwise, a numerical value for the function.

        EXAMPLES::

            sage: hypergeometric._evalf_try_((1.0,), (2.0,), 3.0)
            6.36184564106256
            sage: hypergeometric._evalf_try_((1.0, 1), (), 3.0)
            -0.0377593153441588 + 0.750349833788561*I
            sage: hypergeometric._evalf_try_((1, 1), (), 3)    # exact input
            sage: hypergeometric._evalf_try_((x,), (), 1.0)    # symbolic
            sage: hypergeometric._evalf_try_(1.0, 2.0, 3.0)    # not tuples
        """
        # We need to override this for hypergeometric functions since
        # the first 2 arguments are tuples and the generic _evalf_try_
        # cannot handle that.
        if not isinstance(a,tuple) or not isinstance(b,tuple):
            return None
        args = list(a) + list(b) + [z]
        if any(self._is_numerical(x) for x in args):
            if not any(isinstance(x, Expression) for x in args):
                p = get_coercion_model().common_parent(*args)
                return self._evalf_(a, b, z, parent=p)

            def __truediv__(left, right):
                """
                Return the result of the division of ``left`` by ``right``, if possible.

                This top-level implementation delegates the work to
                the ``_div_`` method if ``left`` and ``right`` have
                the same parent and to coercion otherwise. See the
                extensive documentation at the top of
                :ref:`sage.structure.element`.

                .. SEEALSO:: :meth:`_div_`

                EXAMPLES::

                    sage: G = FreeGroup(2)
                    sage: x0, x1 = G.group_generators()
                    sage: c1 = cartesian_product([x0, x1])
                    sage: c2 = cartesian_product([x1, x0])
                    sage: c1.__div__(c2)
                    (x0*x1^-1, x1*x0^-1)
                    sage: c1 / c2
                    (x0*x1^-1, x1*x0^-1)

                Division supports coercion::

                    sage: C = cartesian_product([G, G])
                    sage: H = Hom(G, C)
                    sage: phi = H(lambda g: cartesian_product([g, g]))
                    sage: phi.register_as_coercion()
                    sage: x1 / c1
                    (x1*x0^-1, 1)
                    sage: c1 / x1
                    (x0*x1^-1, 1)

                Depending on how the division itself is implemented in
                :meth:`_div_`, division may fail even when ``right``
                actually divides ``left``::

                    sage: x = cartesian_product([2, 1])
                    sage: y = cartesian_product([1, 1])
                    sage: x / y
                    (2, 1)
                    sage: x / x
                    Traceback (most recent call last):
                    ...
                    TypeError: no conversion of this rational to integer

                TESTS::

                    sage: c1.__div__.__module__
                    'sage.categories.magmas'
                """
                from sage.structure.element import have_same_parent
                if have_same_parent(left, right):
                    return left._div_(right)
                from sage.structure.element import get_coercion_model
                import operator
                return get_coercion_model().bin_op(left, right, operator.div)

    def direct_sum(self, other):
        """
        Return the direct sum.

        INPUT:

        - ``other`` -- a filtered vector space.

        OUTPUT:

        The direct sum as a filtered vector space.

        EXAMPLES::

            sage: V = FilteredVectorSpace(2, 0)
            sage: W = FilteredVectorSpace({0:[(1,-1),(2,1)], 1:[(1,1)]})
            sage: V.direct_sum(W)
            QQ^4 >= QQ^1 >= 0
            sage: V + W    # syntactic sugar
            QQ^4 >= QQ^1 >= 0
            sage: V + V == FilteredVectorSpace(4, 0)
            True

            sage: W = FilteredVectorSpace([(1,-1),(2,1)], {1:[0,1], 2:[1]})
            sage: V + W
            QQ^4 >= QQ^2 >= QQ^1 >= 0

        A suitable base ring is chosen if they do not match::

            sage: v = [(1,0), (0,1)]
            sage: F1 = FilteredVectorSpace(v, {0:[0], 1:[1]}, base_ring=QQ)
            sage: F2 = FilteredVectorSpace(v, {0:[0], 1:[1]}, base_ring=RDF)
            sage: F1 + F2
            RDF^4 >= RDF^2 >= 0
        """
        from sage.structure.element import get_coercion_model
        base_ring = get_coercion_model().common_parent(self.base_ring(), other.base_ring())
        # construct the generators
        self_gens, self_filt = self.presentation()
        other_gens, other_filt = other.presentation()
        generators = \
            [ list(v) + [base_ring.zero()]*other.dimension() for v in self_gens  ] + \
            [ [base_ring.zero()]*self.dimension() + list(v)  for v in other_gens ]
        # construct the filtration dictionary
        def join_indices(self_indices, other_indices):
            self_indices = tuple(self_indices)
            other_indices = tuple(i + len(self_gens) for i in other_indices)
            return self_indices + other_indices
        filtration = dict()
        self_indices = set()
        other_indices = set()
        for deg in reversed(uniq(self_filt.keys() + other_filt.keys())):
            self_indices.update(self_filt.get(deg, []))
            other_indices.update(other_filt.get(deg, []))
            gens = join_indices(self_indices, other_indices)
            filtration[deg] = gens
        return FilteredVectorSpace(generators, filtration, base_ring=base_ring)

    def __eq__(self, other):
        r"""
        Return if this growth element is equal to ``other``.

        INPUT:

        - ``other`` -- an element.

        OUTPUT:

        A boolean.

        .. NOTE::

            This function uses the coercion model to find a common
            parent for the two operands.

            The comparison of two elements with the same parent is done in
            :meth:`_eq_`.

        EXAMPLES::

            sage: import sage.rings.asymptotic.growth_group as agg
            sage: G = agg.GenericGrowthGroup(ZZ)
            sage: G.an_element() == G.an_element()
            True
            sage: G(raw_element=42) == G(raw_element=7)
            False

        ::

            sage: G_ZZ = agg.GenericGrowthGroup(ZZ)
            sage: G_QQ = agg.GenericGrowthGroup(QQ)
            sage: G_ZZ(raw_element=1) == G_QQ(raw_element=1)
            True

        ::

            sage: P_ZZ = agg.MonomialGrowthGroup(ZZ, 'x')
            sage: P_QQ = agg.MonomialGrowthGroup(QQ, 'x')
            sage: P_ZZ.gen() == P_QQ.gen()
            True
            sage: ~P_ZZ.gen() == P_ZZ.gen()
            False
            sage: ~P_ZZ(1) == P_ZZ(1)
            True
        """
        from sage.structure.element import have_same_parent
        if have_same_parent(self, other):
            return self._eq_(other)

        from sage.structure.element import get_coercion_model
        import operator
        try:
            return get_coercion_model().bin_op(self, other, operator.eq)
        except TypeError:
            return False

        def cup_product(self, cochain):
            """
            Return the cup product with another cochain

            INPUT:

            - ``cochain`` -- cochain over the same cell complex

            EXAMPLES::

                sage: T2 = simplicial_complexes.Torus()
                sage: C1 = T2.n_chains(1, base_ring=ZZ, cochains=True)
                sage: def l(i, j):
                ....:      return C1(Simplex([i, j]))
                sage: l1 = l(1, 3) + l(1, 4) + l(1, 6) + l(2, 4) - l(4, 5) + l(5, 6)
                sage: l2 = l(1, 6) - l(2, 3) - l(2, 5) + l(3, 6) - l(4, 5) + l(5, 6)

            The two one-cocycles are cohomology generators::

                sage: l1.is_cocycle(), l1.is_coboundary()
                (True, False)
                sage: l2.is_cocycle(), l2.is_coboundary()
                (True, False)

            Their cup product is a two-cocycle that is again non-trivial in
            cohomology::

                sage: l12 = l1.cup_product(l2)
                sage: l12
                \chi_(1, 3, 6) - \chi_(2, 4, 5) - \chi_(4, 5, 6)
                sage: l1.parent().degree(), l2.parent().degree(), l12.parent().degree()
                (1, 1, 2)
                sage: l12.is_cocycle(), l12.is_coboundary()
                (True, False)
            """
            if not isinstance(cochain.parent(), Cochains):
                raise ValueError('argument must be a cochain')
            if cochain.parent().cell_complex() != self.parent().cell_complex():
                raise ValueError('cochain must be over the same cell complex')
            left_deg = self.parent().degree()
            right_deg = cochain.parent().degree()
            left_chains = self.parent().dual()
            right_chains = cochain.parent().dual()
            base_ring = get_coercion_model().common_parent(
                left_chains.base_ring(), right_chains.base_ring())
            cx = self.parent().cell_complex()
            codomain = cx.n_chains(
                left_deg + right_deg, base_ring=base_ring, cochains=True)
            accumulator = codomain.zero()
            for cell in codomain.indices():
                for (coeff, left_cell, right_cell) in cx.alexander_whitney(cell, left_deg):
                        if not coeff:
                            continue
                        left = left_chains(left_cell)
                        right = right_chains(right_cell)
                        accumulator += codomain(cell) * coeff * self.eval(left) * cochain.eval(right)
            return accumulator

 def apply_matrix(self,m,in_place=True, mapping=False):
     r"""
     Carry out the GL(2,R) action of m on this surface and return the result.
     
     If in_place=True, then this is done in place and changes the surface. 
     This can only be carried out if the surface is finite and mutable.
     
     If mapping=True, then we return a GL2RMapping between this surface and its image. 
     In this case in_place must be False.
     
     If in_place=False, then a copy is made before the deformation.
     """
     if mapping==True:
         assert in_place==False, "Can not modify in place and return a mapping."
         return GL2RMapping(self, m)
     if not in_place:
         if self.is_finite():
             from sage.structure.element import get_coercion_model
             cm = get_coercion_model()
             field = cm.common_parent(self.base_ring(), m.base_ring())
             s=self.copy(mutable=True, new_field=field)
             return s.apply_matrix(m)
         else:
             return m*self
     else:
         # Make sure m is in the right state
         from sage.matrix.constructor import Matrix
         m=Matrix(self.base_ring(), 2, 2, m)
         assert m.det()!=self.base_ring().zero(), "Can not deform by degenerate matrix."
         assert self.is_finite(), "In place GL(2,R) action only works for finite surfaces."
         us=self.underlying_surface()
         assert us.is_mutable(), "In place changes only work for mutable surfaces."
         for label in self.label_iterator():
             us.change_polygon(label,m*self.polygon(label))
         if m.det()<self.base_ring().zero():
             # Polygons were all reversed orientation. Need to redo gluings.
             
             # First pass record new gluings in a dictionary.
             new_glue={}
             seen_labels=set()
             for p1 in self.label_iterator():
                 n1=self.polygon(p1).num_edges()
                 for e1 in xrange(n1):
                     p2,e2=self.opposite_edge(p1,e1)
                     n2=self.polygon(p2).num_edges()
                     if p2 in seen_labels:
                         pass
                     elif p1==p2 and e1>e2:
                         pass
                     else:
                         new_glue[(p1, n1-1-e1)]=(p2, n2-1-e2)
                 seen_labels.add(p1)
             # Second pass: reassign gluings
             for (p1,e1),(p2,e2) in new_glue.iteritems():
                 us.change_edge_gluing(p1,e1,p2,e2)
         return self

 def __mul__(self,other):
     parent = self.parent()
     if isinstance(other,Matrix_inexact) and parent.base_ring() is other.parent().base_ring():
         group = self.group()
         if other.group() != group:
             raise TypeError("Can multiply only two matrices in the same group")
         return self._matrix_times_matrix_(other)
     else:
         from sage.structure.element import get_coercion_model
         return get_coercion_model().bin_op(self,other,mul)

 def __mul__(self,other):
     parent = self.parent()
     if parent == other.parent():
         group = self.group()
         if other.group() != group:
             raise GroupError("Can multiply only two elements in the same group")
         return self._mul_(other)
     else:
         from sage.structure.element import get_coercion_model
         return get_coercion_model().bin_op(self,other,mul)

        def eval(self, other):
            """
            Evaluate this cochain on the chain ``other``.

            INPUT:

            - ``other`` -- a chain for the same cell complex in the
              same dimension with the same base ring

            OUTPUT: scalar

            EXAMPLES::

                sage: S2 = simplicial_complexes.Sphere(2)
                sage: C_2 = S2.n_chains(1)
                sage: C_2_co = S2.n_chains(1, cochains=True)
                sage: x = C_2.basis()[Simplex((0,2))]
                sage: y = C_2.basis()[Simplex((1,3))]
                sage: z = x+2*y
                sage: a = C_2_co.basis()[Simplex((1,3))]
                sage: b = C_2_co.basis()[Simplex((0,3))]
                sage: c = 3*a-2*b
                sage: z
                (0, 2) + 2*(1, 3)
                sage: c
                -2*\chi_(0, 3) + 3*\chi_(1, 3)
                sage: c.eval(z)
                6

            TESTS::

                sage: z.eval(c) # z is not a cochain
                Traceback (most recent call last):
                ...
                AttributeError: 'Chains_with_category.element_class' object has no attribute 'eval'
                sage: c.eval(c) # can't evaluate a cochain on a cochain
                Traceback (most recent call last):
                ...
                ValueError: argument is not a chain
            """
            if not isinstance(other.parent(), Chains):
                raise ValueError('argument is not a chain')
            if other.parent().indices() != self.parent().indices():
                raise ValueError('the cells are not compatible')
            result = sum(coeff * other.coefficient(cell)
                         for cell, coeff in self)
            R = self.base_ring()
            if R != other.base_ring():
                cm = get_coercion_model()
                R = cm.common_parent(R, other.base_ring())
            return R(result)

        def __rmul__(right, left):
            """
            TESTS::

                sage: F = CombinatorialFreeModule(QQ, ["a", "b"])
                sage: x = F.monomial("a")
                sage: int(2) * x
                2*B['a']

            TODO: make a better unit test once Modules().example() is implemented
            """
            from sage.structure.element import get_coercion_model
            import operator
            return get_coercion_model().bin_op(left, right, operator.mul)

    def tangent_vector(self, lab, p, v, ring=None):
        r"""
        Return a tangent vector.

        INPUT:

        - ``lab`` -- label of a polygon

        - ``p`` -- coordinates of a point in the polygon

        - ``v`` -- coordinates of a vector in R^2
        
        EXAMPLES::

            sage: from flatsurf.geometry.chamanara import chamanara_surface
            sage: S = chamanara_surface(1/2)
            sage: S.tangent_vector(S.base_label(), (1/2,1/2), (1,1))
            SimilaritySurfaceTangentVector in polygon (1, [-1  0]
            [ 0 -1]) based at (1/2, -3/2) with vector (1, 1)
            sage: K.<sqrt2> = QuadraticField(2)
            sage: S.tangent_vector(S.base_label(), (1/2,1/2), (1,sqrt2))
            SimilaritySurfaceTangentVector in polygon (1, [-1  0]
            [ 0 -1]) based at (1/2, -3/2) with vector (1, sqrt2)
        """
        p = vector(p)
        v = vector(v)

        if p.parent().dimension() != 2 or v.parent().dimension() != 2:
            raise ValueError("p (={!r}) and v (={!v}) should have two coordinates")

        if ring is None:
            R = p.base_ring()
            if R != v.base_ring():
                from sage.structure.element import get_coercion_model
                cm = get_coercion_model()
                R = cm.common_parent(R, v.base_ring())
                p = p.change_ring(R)
                v = v.change_ring(R)
    
            R2 = self.base_ring()
            if R != R2:
                if R2.has_coerce_map_from(R):
                    p = p.change_ring(R2)
                    v = v.change_ring(R2)
                    R = R2
                elif not R.has_coerce_map_from(R2):
                    raise ValueError("not able to find a common ring for arguments")
            return self.tangent_bundle(R)(lab, p, v)
        else:
            return self.tangent_bundle(ring)(lab, p, v)