gh-35452: fix all pycodestyle E303 warnings in rings/
    
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### 📚 Description

This is fixing all pep8 E303 warnings in the rings/ folder

Purely done with autopep8.

There remains only to handle later schemes/ in another pull request and
then activate the check in the linter.

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### ⌛ Dependencies

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URL: #35452
Reported by: Frédéric Chapoton
Reviewer(s): Matthias Köppe

Author: Release Manager

src/sage/rings/asymptotic/asymptotic_expansion_generators.py

@@ -197,7 +197,6 @@ def Stirling(var, precision=None, skip_constant_factor=False):
 
         return result
 
-
     @staticmethod
     def log_Stirling(var, precision=None, skip_constant_summand=False):
         r"""
@@ -321,7 +320,6 @@ def log_Stirling(var, precision=None, skip_constant_summand=False):
 
         return result
 
-
     @staticmethod
     def _log_StirlingNegativePowers_(var, precision):
         r"""
@@ -371,7 +369,6 @@ def _log_StirlingNegativePowers_(var, precision):
                      A.zero())
         return result + (1 / n**(2*precision + 1)).O()
 
-
     @staticmethod
     def HarmonicNumber(var, precision=None, skip_constant_summand=False):
         r"""
@@ -477,7 +474,6 @@ def HarmonicNumber(var, precision=None, skip_constant_summand=False):
 
         return result
 
-
     @staticmethod
     def Binomial_kn_over_n(var, k, precision=None, skip_constant_factor=False):
         r"""
@@ -603,7 +599,6 @@ def Binomial_kn_over_n(var, k, precision=None, skip_constant_factor=False):
 
         return result
 
-
     @staticmethod
     def SingularityAnalysis(var, zeta=1, alpha=0, beta=0, delta=0,
                             precision=None, normalized=True):
@@ -961,7 +956,6 @@ def inverse_gamma_derivative(shift, r):
         if precision is None:
             precision = series_precision()
 
-
         if not normalized and not (beta in ZZ and delta in ZZ):
             raise ValueError("beta and delta must be integers")
         if delta != 0:
@@ -1040,7 +1034,6 @@ def inverse_gamma_derivative(shift, r):
 
         return result
 
-
     @staticmethod
     @experimental(20050)
     def ImplicitExpansion(var, phi, tau=None, precision=None):
@@ -1208,7 +1201,6 @@ def ansatz(prec=precision):
 
         return A(tau) + ansatz(prec=precision-1).map_coefficients(lambda term: term.subs(solution_dict).simplify_rational())
 
-
     @staticmethod
     @experimental(20050)
     def ImplicitExpansionPeriodicPart(var, phi, period, tau=None, precision=None):
@@ -1298,7 +1290,6 @@ def ImplicitExpansionPeriodicPart(var, phi, period, tau=None, precision=None):
         Z = aperiodic_expansion.parent().gen()
         return 1/rho * (aperiodic_expansion/(1 - 1/Z))**(1/period)
 
-
     @staticmethod
     def InverseFunctionAnalysis(var, phi, tau=None, period=1, precision=None):
         r"""

src/sage/rings/asymptotic/asymptotic_ring.py

@@ -1209,7 +1209,6 @@ def _mul_term_(self, term):
         return self.parent()(self.summands.mapped(lambda element: term * element),
                              simplify=simplify, convert=False)
 
-
     def _mul_(self, other):
         r"""
         Multiply this asymptotic expansion by another asymptotic expansion ``other``.
@@ -1309,7 +1308,6 @@ def _div_(self, other):
         """
         return self * ~other
 
-
     def __invert__(self, precision=None):
         r"""
         Return the multiplicative inverse of this element.
@@ -1485,7 +1483,6 @@ def exact_part(self):
 
         return self.parent(exact_terms)
 
-
     def error_part(self):
         r"""
         Return the expansion consisting of all error terms of this
@@ -1511,7 +1508,6 @@ def error_part(self):
                     if not term.is_exact()),
                    parent.zero())
 
-
     def __pow__(self, exponent, precision=None):
         r"""
         Calculate the power of this asymptotic expansion to the given ``exponent``.
@@ -1717,10 +1713,8 @@ def __pow__(self, exponent, precision=None):
             raise combine_exceptions(
                 ValueError('Cannot take %s to the exponent %s.' % (self, exponent)), e)
 
-
     pow = __pow__
 
-
     def __pow_number__(self, exponent, precision=None, check_convergence=False):
         r"""
         Return the power of this asymptotic expansion to some
@@ -1873,7 +1867,6 @@ def binomials(a):
 
         return result * pmax
 
-
     def sqrt(self, precision=None):
         r"""
         Return the square root of this asymptotic expansion.
@@ -2244,7 +2237,6 @@ def rpow(self, base, precision=None, locals=None):
         if not expr_o:
             return large_result
 
-
         if base == 'e':
             geom = expr_o
         else:
@@ -3911,7 +3903,6 @@ def _create_empty_summands_():
                             can_merge=can_absorb,
                             merge=absorption)
 
-
     def _create_element_in_extension_(self, term, old_term_parent=None):
         r"""
         Create an element in an extension of this asymptotic ring which

src/sage/rings/asymptotic/asymptotics_multivariate_generating_functions.py

@@ -3324,7 +3324,6 @@ def _an_element_(self):
         from sage.rings.semirings.non_negative_integer_semiring import NN
         return self(NN.an_element(), [(self.base().an_element(), NN(3))])
 
-
     Element = FractionWithFactoredDenominator
 
 

src/sage/rings/asymptotic/growth_group.py

@@ -381,7 +381,6 @@ def __init__(self, var, repr=None, latex_name=None, ignore=None):
             raise ValueError('Variable names %s are not pairwise distinct.' %
                              (var_bases,))
 
-
         self.var_bases = var_bases
         self.var_repr = var_repr
 
@@ -1679,7 +1678,6 @@ class GenericGrowthGroup(UniqueRepresentation, Parent, WithLocals):
     # enable the category framework for elements
     Element = GenericGrowthElement
 
-
     # set everything up to determine category
     from sage.categories.sets_cat import Sets
     from sage.categories.posets import Posets

src/sage/rings/asymptotic/growth_group_cartesian.py

@@ -196,7 +196,6 @@ def create_key_and_extra_args(self, growth_groups, category, **kwds):
 
         return (tuple(growth_groups), category), kwds
 
-
     def create_object(self, version, args, **kwds):
         r"""
         Create an object from the given arguments.
@@ -287,7 +286,6 @@ class GenericProduct(CartesianProductPoset, GenericGrowthGroup):
 
     __classcall__ = CartesianProductPoset.__classcall__
 
-
     def __init__(self, sets, category, **kwds):
         r"""
         See :class:`GenericProduct` for details.
@@ -319,10 +317,8 @@ def __init__(self, sets, category, **kwds):
 
         GenericGrowthGroup.__init__(self, sets[0], Vars, self.category(), **kwds)
 
-
     __hash__ = CartesianProductPoset.__hash__
 
-
     def some_elements(self):
         r"""
         Return some elements of this Cartesian product of growth groups.
@@ -399,7 +395,6 @@ def _create_element_in_extension_(self, element):
                                             category=self.category())
         return parent(element)
 
-
     def _element_constructor_(self, data):
         r"""
         Converts the given object to an element of this Cartesian
@@ -520,10 +515,8 @@ def convert_factors(data, raw_data):
 
         return convert_factors((data,), data)
 
-
     _repr_ = GenericGrowthGroup._repr_
 
-
     def _repr_short_(self):
         r"""
         A short (shorter than :meth:`._repr_`) representation string
@@ -547,7 +540,6 @@ def _repr_short_(self):
         """
         return ' * '.join(S._repr_short_() for S in self.cartesian_factors())
 
-
     def _convert_factors_(self, factors):
         r"""
         Helper method. Try to convert some ``factors`` to an
@@ -617,7 +609,6 @@ def get_factors(data):
         return prod(self.cartesian_injection(*fs)
                     for f in factors for fs in get_factors(f))
 
-
     def cartesian_injection(self, factor, element):
         r"""
         Inject the given element into this Cartesian product at the given factor.
@@ -642,7 +633,6 @@ def cartesian_injection(self, factor, element):
         return self(tuple((f.one() if f != factor else element)
                           for f in self.cartesian_factors()))
 
-
     def _coerce_map_from_(self, S):
         r"""
         Return whether ``S`` coerces into this growth group.
@@ -677,7 +667,6 @@ def _coerce_map_from_(self, S):
                for f in factors):
             return True
 
-
     def _pushout_(self, other):
         r"""
         Construct the pushout of this and the other growth group. This is called by
@@ -864,7 +853,6 @@ def next_custom(self):
         from sage.categories.cartesian_product import cartesian_product
         return pushout(cartesian_product(newS), cartesian_product(newO))
 
-
     def gens_monomial(self):
         r"""
         Return a tuple containing monomial generators of this growth group.
@@ -900,7 +888,6 @@ def gens_monomial(self):
             for factor in self.cartesian_factors()),
                    tuple())
 
-
     def variable_names(self):
         r"""
         Return the names of the variables.
@@ -921,13 +908,11 @@ def variable_names(self):
         from itertools import groupby
         return tuple(v for v, _ in groupby(vars))
 
-
     class Element(CartesianProductPoset.Element):
 
         from .growth_group import _is_lt_one_
         is_lt_one = _is_lt_one_
 
-
         def _repr_(self, latex=False):
             r"""
             A representation string for this Cartesian product element.
@@ -961,7 +946,6 @@ def _repr_(self, latex=False):
                 return '1'
             return s
 
-
         def _latex_(self):
             r"""
             A representation string for this Cartesian product element.
@@ -982,7 +966,6 @@ def _latex_(self):
             """
             return self._repr_(latex=True)
 
-
         def __pow__(self, exponent):
             r"""
             Calculate the power of this growth element to the given
@@ -1013,7 +996,6 @@ def __pow__(self, exponent):
             return self.parent()._create_element_in_extension_(
                 tuple(x ** exponent for x in self.cartesian_factors()))
 
-
         def factors(self):
             r"""
             Return the atomic factors of this growth element. An atomic factor
@@ -1062,12 +1044,10 @@ def factors(self):
                             if not f.is_one()),
                        tuple())
 
-
         from .growth_group import _log_factor_, _log_
         log = _log_
         log_factor = _log_factor_
 
-
         def _log_factor_(self, base=None, locals=None):
             r"""
             Helper method for calculating the logarithm of the factorization
@@ -1120,11 +1100,9 @@ def try_create_growth(g):
                     ArithmeticError('Cannot build log(%s) in %s.' %
                                     (self, self.parent())), e)
 
-
         from .growth_group import _rpow_
         rpow = _rpow_
 
-
         def _rpow_element_(self, base):
             r"""
             Return an element which is the power of ``base`` to this
@@ -1166,7 +1144,6 @@ def _rpow_element_(self, base):
                 raise ValueError  # calling method has to deal with it...
             return factor._rpow_element_(base)
 
-
         def exp(self):
             r"""
             The exponential of this element.
@@ -1211,7 +1188,6 @@ def exp(self):
             """
             return self.rpow('e')
 
-
         def __invert__(self):
             r"""
             Return the multiplicative inverse of this Cartesian product.
@@ -1235,7 +1211,6 @@ def __invert__(self):
             return self.parent()._create_element_in_extension_(
                 tuple(~x for x in self.cartesian_factors()))
 
-
         def _substitute_(self, rules):
             r"""
             Substitute the given ``rules`` in this
@@ -1378,7 +1353,6 @@ def _singularity_analysis_(self, var, zeta, precision):
                     'singularity analysis of {} not yet implemented '
                     'since it has more than two factors'.format(self))
 
-
         def variable_names(self):
             r"""
             Return the names of the variables of this growth element.
@@ -1408,7 +1382,6 @@ def variable_names(self):
             from itertools import groupby
             return tuple(v for v, _ in groupby(vars))
 
-
     CartesianProduct = CartesianProductGrowthGroups
 
 

src/sage/rings/asymptotic/term_monoid.py

@@ -4640,7 +4640,6 @@ def __init__(self, parent, growth, valid_from, **kwds):
                 raise ValueError('B-Term has not defined all variables which occur in the term in valid_from.')
         self.valid_from = valid_from
 
-
     def construction(self):
         r"""
         Return a construction of this term.

src/sage/rings/continued_fraction.py

@@ -1018,8 +1018,6 @@ def __bool__(self):
         """
         return bool(self.quotient(0)) or self.quotient(1) is not Infinity
 
-
-
     def is_zero(self):
         r"""
         Test whether ``self`` is zero.

src/sage/rings/function_field/ideal_polymod.py

@@ -118,8 +118,6 @@ def __bool__(self):
         """
         return self._hnf.nrows() != 0
 
-
-
     def __hash__(self):
         """
         Return the hash of this ideal.