Owaner/CodeComputeX · Datasets at Hugging Face

code stringlengths 17 6.64M
class ArithmeticSubgroup_Permutation_class(ArithmeticSubgroup): "\n A subgroup of `\\SL_2(\\ZZ)` defined by the action of generators on its\n coset graph.\n\n The class stores the action of generators `s_2, s_3, l, r` on right cosets\n `Hg` of a finite index subgroup `H < \\SL_2(\\ZZ)`. In particular ...
class OddArithmeticSubgroup_Permutation(ArithmeticSubgroup_Permutation_class): '\n An arithmetic subgroup of `\\SL(2, \\ZZ)` not containing `-1`,\n represented in terms of the right action of `\\SL(2, \\ZZ)` on its\n cosets.\n\n EXAMPLES::\n\n sage: G = ArithmeticSubgroup_Permutation(S2="(1,2,3...
class EvenArithmeticSubgroup_Permutation(ArithmeticSubgroup_Permutation_class): '\n An arithmetic subgroup of `\\SL(2, \\ZZ)` containing `-1`, represented\n in terms of the right action of `\\SL(2, \\ZZ)` on its cosets.\n\n EXAMPLES:\n\n Construct a noncongruence subgroup of index 7 (the smallest poss...
def HsuExample10(): '\n An example of an index 10 arithmetic subgroup studied by Tim Hsu.\n\n EXAMPLES::\n\n sage: import sage.modular.arithgroup.arithgroup_perm as ap\n sage: ap.HsuExample10()\n Arithmetic subgroup with permutations of right cosets\n S2=(1,2)(3,4)(5,6)(7,8)(9,1...
def HsuExample18(): '\n An example of an index 18 arithmetic subgroup studied by Tim Hsu.\n\n EXAMPLES::\n\n sage: import sage.modular.arithgroup.arithgroup_perm as ap\n sage: ap.HsuExample18()\n Arithmetic subgroup with permutations of right cosets\n S2=(1,5)(2,11)(3,10)(4,15)(...
def Gamma_constructor(N): '\n Return the congruence subgroup `\\Gamma(N)`.\n\n EXAMPLES::\n\n sage: Gamma(5) # indirect doctest\n Congruence Subgroup Gamma(5)\n sage: G = Gamma(23)\n sage: G is Gamma(23)\n True\n sage: TestSuite(G).run()\n\n Test global uniquenes...
@richcmp_method class Gamma_class(CongruenceSubgroup): '\n The principal congruence subgroup `\\Gamma(N)`.\n ' def _repr_(self): "\n Return the string representation of self.\n\n EXAMPLES::\n\n sage: Gamma(133)._repr_()\n 'Congruence Subgroup Gamma(133)'\n ...
def is_Gamma(x): '\n Return True if x is a congruence subgroup of type Gamma.\n\n EXAMPLES::\n\n sage: from sage.modular.arithgroup.all import is_Gamma\n sage: is_Gamma(Gamma0(13))\n False\n sage: is_Gamma(Gamma(4))\n True\n ' return isinstance(x, Gamma_class)
def _lift_pair(U, V, N): "\n Utility function. Given integers ``U, V, N``, with `N \\ge 1` and `{\\rm\n gcd}(U, V, N) = 1`, return a pair `(u, v)` congruent to `(U, V) \\bmod N`,\n such that `{\\rm gcd}(u,v) = 1`, `u, v \\ge 0`, `v` is as small as possible,\n and `u` is as small as possible for that `...
def is_Gamma0(x): '\n Return True if x is a congruence subgroup of type Gamma0.\n\n EXAMPLES::\n\n sage: from sage.modular.arithgroup.all import is_Gamma0\n sage: is_Gamma0(SL2Z)\n True\n sage: is_Gamma0(Gamma0(13))\n True\n sage: is_Gamma0(Gamma1(6))\n False...
def Gamma0_constructor(N): '\n Return the congruence subgroup Gamma0(N).\n\n EXAMPLES::\n\n sage: G = Gamma0(51) ; G # indirect doctest\n Congruence Subgroup Gamma0(51)\n sage: G == Gamma0(51)\n True\n sage: G is Gamma0(51)\n True\n ' from .all import SL2Z ...
class Gamma0_class(GammaH_class): "\n The congruence subgroup `\\Gamma_0(N)`.\n\n TESTS::\n\n sage: Gamma0(11).dimension_cusp_forms(2)\n 1\n sage: a = Gamma0(1).dimension_cusp_forms(2); a\n 0\n sage: type(a)\n <class 'sage.rings.integer.Integer'>\n sage: Gamm...
def is_Gamma1(x): '\n Return True if x is a congruence subgroup of type Gamma1.\n\n EXAMPLES::\n\n sage: from sage.modular.arithgroup.all import is_Gamma1\n sage: is_Gamma1(SL2Z)\n False\n sage: is_Gamma1(Gamma1(13))\n True\n sage: is_Gamma1(Gamma0(6))\n Fals...
def Gamma1_constructor(N): '\n Return the congruence subgroup `\\Gamma_1(N)`.\n\n EXAMPLES::\n\n sage: Gamma1(5) # indirect doctest\n Congruence Subgroup Gamma1(5)\n sage: G = Gamma1(23)\n sage: G is Gamma1(23)\n True\n sage: G is GammaH(23, [1])\n True\n ...
class Gamma1_class(GammaH_class): '\n The congruence subgroup `\\Gamma_1(N)`.\n\n TESTS::\n\n sage: [Gamma1(n).genus() for n in prime_range(2,100)]\n [0, 0, 0, 0, 1, 2, 5, 7, 12, 22, 26, 40, 51, 57, 70, 92, 117, 126, 155, 176, 187, 222, 247, 287, 345]\n sage: [Gamma1(n).index() for n in...
def GammaH_constructor(level, H): '\n Return the congruence subgroup `\\Gamma_H(N)`, which is the subgroup of\n `SL_2(\\ZZ)` consisting of matrices of the form `\\begin{pmatrix} a & b \\\\\n c & d \\end{pmatrix}` with `N \| c` and `a, d \\in H`, for `H` a specified\n subgroup of `(\\ZZ/N\\ZZ)^\\times`....
def is_GammaH(x): '\n Return True if x is a congruence subgroup of type GammaH.\n\n EXAMPLES::\n\n sage: from sage.modular.arithgroup.all import is_GammaH\n sage: is_GammaH(GammaH(13, [2]))\n True\n sage: is_GammaH(Gamma0(6))\n True\n sage: is_GammaH(Gamma1(6))\n ...
def _normalize_H(H, level): '\n Normalize representatives for a given subgroup H of the units\n modulo level.\n\n .. NOTE::\n\n This function does not make any attempt to find a minimal\n set of generators for H. It simply normalizes the inputs for use\n in hashing.\n\n EXAMPLES...
@richcmp_method class GammaH_class(CongruenceSubgroup): '\n The congruence subgroup `\\Gamma_H(N)` for some subgroup `H \\trianglelefteq\n (\\ZZ / N\\ZZ)^\\times`, which is the subgroup of `\\SL_2(\\ZZ)` consisting of\n matrices of the form `\\begin{pmatrix} a &\n b \\\\ c & d \\end{pmatrix}` with `N ...
def _list_subgroup(N, gens): '\n Given an integer ``N`` and a list of integers ``gens``, return a list of\n the elements of the subgroup of `(\\ZZ / N\\ZZ)^\\times` generated by the\n elements of ``gens``.\n\n EXAMPLES::\n\n sage: sage.modular.arithgroup.congroup_gammaH._list_subgroup(11, [3])\...
def _GammaH_coset_helper(N, H): '\n Return a list of coset representatives for H in (Z / NZ)^*.\n\n EXAMPLES::\n\n sage: from sage.modular.arithgroup.congroup_gammaH import _GammaH_coset_helper\n sage: _GammaH_coset_helper(108, [1, 107])\n [1, 5, 7, 11, 13, 17, 19, 23, 25, 29, 31, 35, 3...
def mumu(N): '\n Return 0 if any cube divides `N`. Otherwise return\n `(-2)^v` where `v` is the number of primes that\n exactly divide `N`.\n\n This is similar to the Möbius function.\n\n INPUT:\n\n - ``N`` -- an integer at least 1\n\n OUTPUT: Integer\n\n EXAMPLES::\n\n sage: from s...
def CongruenceSubgroup_constructor(*args): '\n Attempt to create a congruence subgroup from the given data.\n\n The allowed inputs are as follows:\n\n - A :class:`~sage.groups.matrix_gps.matrix_group.MatrixGroup` object. This\n must be a group of matrices over `\\ZZ / N\\ZZ` for some `N`, with\n ...
def is_CongruenceSubgroup(x): '\n Return True if x is of type CongruenceSubgroup.\n\n Note that this may be False even if `x` really is a congruence subgroup --\n it tests whether `x` is "obviously" congruence, i.e.~whether it has a\n congruence subgroup datatype. To test whether or not an arithmetic ...
class CongruenceSubgroupBase(ArithmeticSubgroup): def __init__(self, level): '\n Create a congruence subgroup with given level.\n\n EXAMPLES::\n\n sage: Gamma0(500)\n Congruence Subgroup Gamma0(500)\n ' level = ZZ(level) if (level <= 0): ...
class CongruenceSubgroupFromGroup(CongruenceSubgroupBase): '\n A congruence subgroup, defined by the data of an integer `N` and a subgroup\n `G` of the finite group `SL(2, \\ZZ / N\\ZZ)`; the congruence subgroup\n consists of all the matrices in `SL(2, \\ZZ)` whose reduction modulo `N`\n lies in `G`.\...
class CongruenceSubgroup(CongruenceSubgroupFromGroup): '\n One of the "standard" congruence subgroups `\\Gamma_0(N)`, `\\Gamma_1(N)`,\n `\\Gamma(N)`, or `\\Gamma_H(N)` (for some `H`).\n\n This class is not intended to be instantiated directly. Derived subclasses\n must override ``_contains_sl2``, ``_r...
def _minimize_level(G): '\n Utility function. Given a matrix group `G` contained in `SL(2, \\ZZ / N\\ZZ)`\n for some `N`, test whether or not `G` is the preimage of a subgroup of\n smaller level, and if so, return that subgroup.\n\n The trivial group is handled specially: instead of returning a group,...
def is_SL2Z(x): '\n Return True if x is the modular group `\\SL_2(\\ZZ)`.\n\n EXAMPLES::\n\n sage: from sage.modular.arithgroup.all import is_SL2Z\n sage: is_SL2Z(SL2Z)\n True\n sage: is_SL2Z(Gamma0(6))\n False\n ' return isinstance(x, SL2Z_class)
class SL2Z_class(Gamma0_class): '\n The full modular group `\\SL_2(\\ZZ)`, regarded as a congruence\n subgroup of itself.\n ' def __init__(self): '\n The modular group `\\SL_2(\\Z)`.\n\n EXAMPLES::\n\n sage: G = SL2Z; G\n Modular Group SL(2,Z)\n ...
def _SL2Z_ref(): '\n Return SL2Z. (Used for pickling SL2Z.)\n\n EXAMPLES::\n\n sage: sage.modular.arithgroup.congroup_sl2z._SL2Z_ref()\n Modular Group SL(2,Z)\n sage: sage.modular.arithgroup.congroup_sl2z._SL2Z_ref() is SL2Z\n True\n ' return SL2Z
def random_even_arithgroup(index, nu2_max=None, nu3_max=None): '\n Return a random even arithmetic subgroup.\n\n EXAMPLES::\n\n sage: import sage.modular.arithgroup.tests as tests\n sage: G = tests.random_even_arithgroup(30); G # random\n Arithmetic subgroup of index 30\n sage: G...
def random_odd_arithgroup(index, nu3_max=None): '\n Return a random odd arithmetic subgroup.\n\n EXAMPLES::\n\n sage: from sage.modular.arithgroup.tests import random_odd_arithgroup\n sage: G = random_odd_arithgroup(20); G #random\n Arithmetic subgroup of index 20\n sage: G.is_od...
class Test(): '\n Testing class for arithmetic subgroup implemented via permutations.\n ' def __init__(self, index=20, index_max=50, odd_probability=0.5): '\n Create an arithmetic subgroup testing object.\n\n INPUT:\n\n - ``index`` - the index of random subgroup to test\n\n...
class DoubleCosetReduction(SageObject): "\n Edges in the Bruhat-Tits tree are represented by cosets of\n matrices in `GL_2`. Given a matrix `x` in `GL_2`, this\n class computes and stores the data corresponding to the\n double coset representation of `x` in terms of a fundamental\n domain of edges ...
class BruhatTitsTree(SageObject, UniqueRepresentation): '\n An implementation of the Bruhat-Tits tree for `GL_2(\\QQ_p)`.\n\n INPUT:\n\n - ``p`` - a prime number. The corresponding tree is then `p+1` regular\n\n EXAMPLES:\n\n We create the tree for `GL_2(\\QQ_5)`::\n\n sage: from sage.modula...
class Vertex(SageObject): '\n This is a structure to represent vertices of quotients of the\n Bruhat-Tits tree. It is useful to enrich the representation of\n the vertex as a matrix with extra data.\n\n INPUT:\n\n - ``p`` - a prime integer.\n\n - ``label`` - An integer which uniquely identifies...
class Edge(SageObject): '\n This is a structure to represent edges of quotients of the\n Bruhat-Tits tree. It is useful to enrich the representation of an\n edge as a matrix with extra data.\n\n INPUT:\n\n - ``p`` - a prime integer.\n\n - ``label`` - An integer which uniquely identifies this edg...
class BruhatTitsQuotient(SageObject, UniqueRepresentation): "\n This function computes the quotient of the Bruhat-Tits tree\n by an arithmetic quaternionic group. The group in question is the\n group of norm 1 elements in an Eichler `\\ZZ[1/p]`-order of some (tame)\n level inside of a definite quatern...
class _btquot_adjuster(Sigma0ActionAdjuster): '\n Callable object that turns matrices into 4-tuples.\n\n Since the modular symbol and harmonic cocycle code use different\n conventions for group actions, this function is used to make sure\n that actions are correct for harmonic cocycle computations.\n\...
def eval_dist_at_powseries(phi, f): '\n Evaluate a distribution on a powerseries.\n\n A distribution is an element in the dual of the Tate ring. The\n elements of coefficient modules of overconvergent modular symbols\n and overconvergent `p`-adic automorphic forms give examples of\n distributions i...
class BruhatTitsHarmonicCocycleElement(HeckeModuleElement): '\n `\\Gamma`-invariant harmonic cocycles on the Bruhat-Tits\n tree. `\\Gamma`-invariance is necessary so that the cocycle can be\n stored in terms of a finite amount of data.\n\n More precisely, given a ``BruhatTitsQuotient`` `T`, harmonic c...
class BruhatTitsHarmonicCocycles(AmbientHeckeModule, UniqueRepresentation): '\n Ensure unique representation\n\n EXAMPLES::\n\n sage: X = BruhatTitsQuotient(3,5)\n sage: M1 = X.harmonic_cocycles( 2, prec = 10)\n sage: M2 = X.harmonic_cocycles( 2, 10)\n sage: M1 is M2\n Tru...
class pAdicAutomorphicFormElement(ModuleElement): "\n Rudimentary implementation of a class for a `p`-adic\n automorphic form on a definite quaternion algebra over `\\QQ`. These\n are required in order to compute moments of measures associated to\n harmonic cocycles on the Bruhat-Tits tree using the o...
class pAdicAutomorphicForms(Module, UniqueRepresentation): Element = pAdicAutomorphicFormElement @staticmethod def __classcall__(cls, domain, U, prec=None, t=None, R=None, overconvergent=False): '\n The module of (quaternionic) `p`-adic automorphic forms.\n\n INPUT:\n\n - ``d...
def gp(): '\n Return a copy of the GP interpreter with the appropriate files loaded.\n\n EXAMPLES::\n\n sage: import sage.modular.buzzard\n sage: sage.modular.buzzard.gp()\n PARI/GP interpreter\n ' global _gp if (_gp is None): _gp = Gp(script_subdirectory='buzzard') ...
def buzzard_tpslopes(p, N, kmax): "\n Return a vector of length kmax, whose `k`'th entry\n (`0 \\leq k \\leq k_{max}`) is the conjectural sequence\n of valuations of eigenvalues of `T_p` on forms of level\n `N`, weight `k`, and trivial character.\n\n This conjecture is due to Kevin Buzzard, and is ...
class Cusp(Element): '\n A cusp.\n\n A cusp is either a rational number or infinity, i.e., an element of\n the projective line over Q. A Cusp is stored as a pair (a,b), where\n gcd(a,b)=1 and a,b are of type Integer.\n\n EXAMPLES::\n\n sage: a = Cusp(2/3); b = Cusp(oo)\n sage: a.paren...
class Cusps_class(Singleton, Parent): '\n The set of cusps.\n\n EXAMPLES::\n\n sage: C = Cusps; C\n Set P^1(QQ) of all cusps\n sage: loads(C.dumps()) == C\n True\n ' def __init__(self): '\n The set of cusps, i.e. `\\mathbb{P}^1(\\QQ)`.\n\n EXAMPLES::...
@cached_function def list_of_representatives(N): "\n Return a list of ideals, coprime to the ideal ``N``, representatives of\n the ideal classes of the corresponding number field.\n\n .. NOTE::\n\n This list, used every time we check `\\Gamma_0(N)` - equivalence of\n cusps, is cached.\n\n ...
@cached_function def NFCusps(number_field): "\n The set of cusps of a number field `K`, i.e. `\\mathbb{P}^1(K)`.\n\n INPUT:\n\n - ``number_field`` -- a number field\n\n OUTPUT:\n\n The set of cusps over the given number field.\n\n EXAMPLES::\n\n sage: x = polygen(ZZ, 'x')\n sage: k...
class NFCuspsSpace(UniqueRepresentation, Parent): "\n The set of cusps of a number field. See ``NFCusps`` for full documentation.\n\n EXAMPLES::\n\n sage: x = polygen(ZZ, 'x')\n sage: k.<a> = NumberField(x^2 + 5)\n sage: kCusps = NFCusps(k); kCusps\n Set of all cusps of Number Fi...
class NFCusp(Element): '\n Create a number field cusp, i.e., an element of `\\mathbb{P}^1(k)`.\n\n A cusp on a number field is either an element of the field or infinity,\n i.e., an element of the projective line over the number field. It is\n stored as a pair (a,b), where a, b are integral elements ...
def Gamma0_NFCusps(N): "\n Return a list of inequivalent cusps for `\\Gamma_0(N)`, i.e., a set of\n representatives for the orbits of ``self`` on `\\mathbb{P}^1(k)`.\n\n INPUT:\n\n - ``N`` -- an integral ideal of the number field k (the level).\n\n OUTPUT:\n\n A list of inequivalent number field...
def number_of_Gamma0_NFCusps(N): "\n Return the total number of orbits of cusps under the action of the\n congruence subgroup `\\Gamma_0(N)`.\n\n INPUT:\n\n - ``N`` -- a number field ideal.\n\n OUTPUT:\n\n integer -- the number of orbits of cusps under Gamma0(N)-action.\n\n EXAMPLES::\n\n ...
def NFCusps_ideal_reps_for_levelN(N, nlists=1): "\n Return a list of lists (``nlists`` different lists) of prime ideals,\n coprime to ``N``, representing every ideal class of the number field.\n\n INPUT:\n\n - ``N`` -- number field ideal.\n\n - ``nlists`` -- optional (default 1). The number of list...
def units_mod_ideal(I): "\n Return integral elements of the number field representing the images of\n the global units modulo the ideal ``I``.\n\n INPUT:\n\n - ``I`` -- number field ideal.\n\n OUTPUT:\n\n A list of integral elements of the number field representing the images of\n the global ...
def eisen(p): '\n Return the Eisenstein number `n` which is the numerator of `(p-1)/12`.\n\n INPUT:\n\n - ``p`` -- a prime\n\n OUTPUT: Integer\n\n EXAMPLES::\n\n sage: [(p, sage.modular.dims.eisen(p)) for p in prime_range(24)]\n [(2, 1), (3, 1), (5, 1), (7, 1), (11, 5), (13, 1), (17, ...
def CO_delta(r, p, N, eps): '\n This is used as an intermediate value in computations related to\n the paper of Cohen-Oesterlé.\n\n INPUT:\n\n - ``r`` -- positive integer\n\n - ``p`` -- a prime\n\n - ``N`` -- positive integer\n\n - ``eps`` -- character\n\n OUTPUT: element of the base r...
def CO_nu(r, p, N, eps): '\n This is used as an intermediate value in computations related to\n the paper of Cohen-Oesterlé.\n\n INPUT:\n\n - ``r`` -- positive integer\n\n - ``p`` -- a prime\n\n - ``N`` -- positive integer\n\n - ``eps`` -- character\n\n OUTPUT: element of the base ring...
def CohenOesterle(eps, k): '\n Compute the Cohen-Oesterlé function associate to eps, `k`.\n\n This is a summand in the formula for the dimension of the space of\n cusp forms of weight `2` with character `\\varepsilon`.\n\n INPUT:\n\n - ``eps`` -- Dirichlet character\n\n - ``k`` -- integer\n\n ...
def dimension_new_cusp_forms(X, k=2, p=0): '\n Return the dimension of the new (or `p`-new) subspace of\n cusp forms for the character or group `X`.\n\n INPUT:\n\n - ``X`` -- integer, congruence subgroup or Dirichlet\n character\n\n - ``k`` -- weight (integer)\n\n - ``p`` -- 0 or a prim...
def dimension_cusp_forms(X, k=2): '\n The dimension of the space of cusp forms for the given congruence\n subgroup or Dirichlet character.\n\n INPUT:\n\n - ``X`` -- congruence subgroup or Dirichlet character\n or integer\n\n - ``k`` -- weight (integer)\n\n EXAMPLES::\n\n sage: fro...
def dimension_eis(X, k=2): '\n The dimension of the space of Eisenstein series for the given\n congruence subgroup.\n\n INPUT:\n\n - ``X`` -- congruence subgroup or Dirichlet character\n or integer\n\n - ``k`` -- weight (integer)\n\n EXAMPLES::\n\n sage: from sage.modular.dims imp...
def dimension_modular_forms(X, k=2): '\n The dimension of the space of cusp forms for the given congruence\n subgroup (either `\\Gamma_0(N)`, `\\Gamma_1(N)`, or\n `\\Gamma_H(N)`) or Dirichlet character.\n\n INPUT:\n\n - ``X`` -- congruence subgroup or Dirichlet character\n\n - ``k`` -- weight ...
def sturm_bound(level, weight=2): '\n Return the Sturm bound for modular forms with given level and weight.\n\n For more details, see the documentation for the ``sturm_bound`` method\n of :class:`sage.modular.arithgroup.CongruenceSubgroup` objects.\n\n INPUT:\n\n - ``level`` -- an integer (interpre...
def trivial_character(N, base_ring=QQ): '\n Return the trivial character of the given modulus, with values in the given\n base ring.\n\n EXAMPLES::\n\n sage: t = trivial_character(7)\n sage: [t(x) for x in [0..20]]\n [0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1]\n ...
def kronecker_character(d): '\n Return the quadratic Dirichlet character (d/.) of minimal\n conductor.\n\n EXAMPLES::\n\n sage: kronecker_character(97389997^2)\n Dirichlet character modulo 37733 of conductor 37733 mapping 1557 \|--> -1, 37346 \|--> -1\n\n ::\n\n sage: a = kronecke...
def kronecker_character_upside_down(d): '\n Return the quadratic Dirichlet character (./d) of conductor d, for\n d > 0.\n\n EXAMPLES::\n\n sage: kronecker_character_upside_down(97389997^2)\n Dirichlet character modulo 37506941597 of conductor 37733 mapping 13533432536 \|--> -1, 22369178537...
def is_DirichletCharacter(x) -> bool: '\n Return ``True`` if ``x`` is of type ``DirichletCharacter``.\n\n EXAMPLES::\n\n sage: from sage.modular.dirichlet import is_DirichletCharacter\n sage: is_DirichletCharacter(trivial_character(3))\n True\n sage: is_DirichletCharacter([1])\n ...
class DirichletCharacter(MultiplicativeGroupElement): '\n A Dirichlet character.\n ' def __init__(self, parent, x, check=True): '\n Create a Dirichlet character with specified values on\n generators of `(\\ZZ/n\\ZZ)^*`.\n\n INPUT:\n\n - ``parent`` -- :class:`Dirichle...
class DirichletGroupFactory(UniqueFactory): '\n Construct a group of Dirichlet characters modulo `N`.\n\n INPUT:\n\n - ``N`` -- positive integer\n\n - ``base_ring`` -- commutative ring; the value ring for the\n characters in this group (default: the cyclotomic field\n `\\QQ(\\zeta_n)`, where...
def is_DirichletGroup(x): '\n Return ``True`` if ``x`` is a Dirichlet group.\n\n EXAMPLES::\n\n sage: from sage.modular.dirichlet import is_DirichletGroup\n sage: is_DirichletGroup(DirichletGroup(11))\n True\n sage: is_DirichletGroup(11)\n False\n sage: is_Dirichlet...
class DirichletGroup_class(WithEqualityById, Parent): '\n Group of Dirichlet characters modulo `N` with values in a ring `R`.\n ' Element = DirichletCharacter def __init__(self, base_ring, modulus, zeta, zeta_order): '\n Create a Dirichlet group.\n\n Not to be called directly ...
def EtaGroup(level): '\n Create the group of eta products of the given level.\n\n EXAMPLES::\n\n sage: EtaGroup(12)\n Group of eta products on X_0(12)\n sage: EtaGroup(1/2)\n Traceback (most recent call last):\n ...\n TypeError: Level (=1/2) must be a positive integ...
class EtaGroupElement(Element): def __init__(self, parent, rdict): '\n Create an eta product object. Usually called implicitly via\n EtaGroup_class.__call__ or the EtaProduct factory function.\n\n EXAMPLES::\n\n sage: EtaProduct(8, {1:24, 2:-24})\n Eta product o...
class EtaGroup_class(UniqueRepresentation, Parent): '\n The group of eta products of a given level under multiplication.\n\n TESTS::\n\n sage: TestSuite(EtaGroup(12)).run()\n\n sage: EtaGroup(12) == EtaGroup(12)\n True\n sage: EtaGroup(12) == EtaGroup(13)\n False\n\n ...
def EtaProduct(level, dic) -> EtaGroupElement: "\n Create an :class:`EtaGroupElement` object representing the function\n `\\prod_{d \| N} \\eta(q^d)^{r_d}`.\n\n This checks the criteria of Ligozat to ensure that this product\n really is the `q`-expansion of a meromorphic function on `X_0(N)`.\n\n IN...
def num_cusps_of_width(N, d) -> Integer: '\n Return the number of cusps on `X_0(N)` of width ``d``.\n\n INPUT:\n\n - ``N`` -- (integer): the level\n\n - ``d`` -- (integer): an integer dividing N, the cusp width\n\n EXAMPLES::\n\n sage: from sage.modular.etaproducts import num_cusps_of_widt...
def AllCusps(N): '\n Return a list of CuspFamily objects corresponding to the cusps of\n `X_0(N)`.\n\n INPUT:\n\n - ``N`` -- (integer): the level\n\n EXAMPLES::\n\n sage: AllCusps(18)\n [(Inf), (c_{2}), (c_{3,1}), (c_{3,2}), (c_{6,1}), (c_{6,2}), (c_{9}), (0)]\n sage: AllCusps...
@richcmp_method class CuspFamily(SageObject): '\n A family of elliptic curves parametrising a region of `X_0(N)`.\n ' def __init__(self, N, width, label=None): "\n Create the cusp of width d on X_0(N) corresponding to the family\n of Tate curves `(\\CC_p/q^d, \\langle \\zeta q\\ra...
def qexp_eta(ps_ring, prec): "\n Return the q-expansion of `\\eta(q) / q^{1/24}`.\n\n Here `\\eta(q)` is Dedekind's function\n\n .. MATH::\n\n \\eta(q) = q^{1/24}\\prod_{n=1}^\\infty (1-q^n).\n\n The result is an element of ``ps_ring``, with precision ``prec``.\n\n INPUT:\n\n - ``ps_ring...
def eta_poly_relations(eta_elements, degree, labels=['x1', 'x2'], verbose=False): "\n Find polynomial relations between eta products.\n\n INPUT:\n\n - ``eta_elements`` - (list): a list of EtaGroupElement objects.\n Not implemented unless this list has precisely two elements. degree\n\n - ``degree...
def _eta_relations_helper(eta1, eta2, degree, qexp_terms, labels, verbose): "\n Helper function used by eta_poly_relations. Finds a basis for the\n space of linear relations between the first qexp_terms of the\n `q`-expansions of the monomials\n `\\eta_1^i * \\eta_2^j` for `0 \\le i,j < degree`,\n ...
def is_HeckeAlgebra(x): '\n Return ``True`` if x is of type HeckeAlgebra.\n\n EXAMPLES::\n\n sage: from sage.modular.hecke.algebra import is_HeckeAlgebra\n sage: is_HeckeAlgebra(CuspForms(1, 12).anemic_hecke_algebra())\n True\n sage: is_HeckeAlgebra(ZZ)\n False\n ' ...
def _heckebasis(M): '\n Return a basis of the Hecke algebra of M as a ZZ-module.\n\n INPUT:\n\n - ``M`` -- a Hecke module\n\n OUTPUT:\n\n a list of Hecke algebra elements represented as matrices\n\n EXAMPLES::\n\n sage: M = ModularSymbols(11,2,1)\n sage: sage.modular.hecke.algebra....
@richcmp_method class HeckeAlgebra_base(CachedRepresentation, CommutativeAlgebra): '\n Base class for algebras of Hecke operators on a fixed Hecke module.\n\n INPUT:\n\n - ``M`` - a Hecke module\n\n EXAMPLES::\n\n sage: CuspForms(1, 12).hecke_algebra() # indirect doctest\n Full Hecke al...
class HeckeAlgebra_full(HeckeAlgebra_base): '\n A full Hecke algebra (including the operators `T_n` where `n` is not\n assumed to be coprime to the level).\n ' def _repr_(self): "\n String representation of self.\n\n EXAMPLES::\n\n sage: ModularForms(37).hecke_algebr...
class HeckeAlgebra_anemic(HeckeAlgebra_base): '\n An anemic Hecke algebra, generated by Hecke operators with index coprime to the level.\n ' def _repr_(self): '\n EXAMPLES::\n\n sage: H = CuspForms(3, 12).anemic_hecke_algebra()._repr_()\n ' return ('Anemic Hecke...
def is_AmbientHeckeModule(x) -> bool: '\n Return ``True`` if ``x`` is of type ``AmbientHeckeModule``.\n\n EXAMPLES::\n\n sage: from sage.modular.hecke.ambient_module import is_AmbientHeckeModule\n sage: is_AmbientHeckeModule(ModularSymbols(6))\n True\n sage: is_AmbientHeckeModule...
class AmbientHeckeModule(module.HeckeModule_free_module): '\n An ambient Hecke module, i.e. a Hecke module that is isomorphic as a module\n over its base ring `R` to the standard free module `R^k` for some `k`. This\n is the base class for ambient spaces of modular forms and modular symbols,\n and for...
class DegeneracyMap(morphism.HeckeModuleMorphism_matrix): '\n A degeneracy map between Hecke modules of different levels.\n\n EXAMPLES:\n\n We construct a number of degeneracy maps::\n\n sage: M = ModularSymbols(33)\n sage: d = M.degeneracy_map(11)\n sage: d\n Hecke module mor...
def is_HeckeModuleElement(x): '\n Return ``True`` if x is a Hecke module element, i.e., of type HeckeModuleElement.\n\n EXAMPLES::\n\n sage: sage.modular.hecke.all.is_HeckeModuleElement(0)\n False\n sage: sage.modular.hecke.all.is_HeckeModuleElement(BrandtModule(37)([1,2,3]))\n T...
class HeckeModuleElement(ModuleElement): '\n Element of a Hecke module.\n ' def __init__(self, parent, x=None): "\n INPUT:\n\n - ``parent`` -- a Hecke module\n\n - ``x`` -- element of the free module associated to parent\n\n EXAMPLES::\n\n sage: v = sage.m...
def is_HeckeOperator(x): '\n Return ``True`` if x is of type HeckeOperator.\n\n EXAMPLES::\n\n sage: from sage.modular.hecke.hecke_operator import is_HeckeOperator\n sage: M = ModularSymbols(Gamma0(7), 4)\n sage: is_HeckeOperator(M.T(3))\n True\n sage: is_HeckeOperator(M.T...
def is_HeckeAlgebraElement(x): '\n Return ``True`` if x is of type HeckeAlgebraElement.\n\n EXAMPLES::\n\n sage: from sage.modular.hecke.hecke_operator import is_HeckeAlgebraElement\n sage: M = ModularSymbols(Gamma0(7), 4)\n sage: is_HeckeAlgebraElement(M.T(3))\n True\n sa...
class HeckeAlgebraElement(AlgebraElement): '\n Base class for elements of Hecke algebras.\n ' def __init__(self, parent): "\n Create an element of a Hecke algebra.\n\n EXAMPLES::\n\n sage: R = ModularForms(Gamma0(7), 4).hecke_algebra()\n sage: sage.modular.he...
class HeckeAlgebraElement_matrix(HeckeAlgebraElement): '\n An element of the Hecke algebra represented by a matrix.\n ' def __init__(self, parent, A): "\n Initialise an element from a matrix. This must be over the base ring\n of self and have the right size.\n\n This is a...
class DiamondBracketOperator(HeckeAlgebraElement_matrix): '\n The diamond bracket operator `\\langle d \\rangle` for some `d \\in \\ZZ /\n N\\ZZ` (which need not be a unit, although if it is not, the operator will\n be zero).\n ' def __init__(self, parent, d): "\n Standard init fun...
class HeckeOperator(HeckeAlgebraElement): '\n The Hecke operator `T_n` for some `n` (which need not be coprime to the\n level). The matrix is not computed until it is needed.\n ' def __init__(self, parent, n): '\n EXAMPLES::\n\n sage: M = ModularSymbols(11)\n sag...

class ArithmeticSubgroup_Permutation_class(ArithmeticSubgroup): "\n A subgroup of `\\SL_2(\\ZZ)` defined by the action of generators on its\n coset graph.\n\n The class stores the action of generators `s_2, s_3, l, r` on right cosets\n `Hg` of a finite index subgroup `H < \\SL_2(\\ZZ)`. In particular ...

class OddArithmeticSubgroup_Permutation(ArithmeticSubgroup_Permutation_class): '\n An arithmetic subgroup of `\\SL(2, \\ZZ)` not containing `-1`,\n represented in terms of the right action of `\\SL(2, \\ZZ)` on its\n cosets.\n\n EXAMPLES::\n\n sage: G = ArithmeticSubgroup_Permutation(S2="(1,2,3...

class EvenArithmeticSubgroup_Permutation(ArithmeticSubgroup_Permutation_class): '\n An arithmetic subgroup of `\\SL(2, \\ZZ)` containing `-1`, represented\n in terms of the right action of `\\SL(2, \\ZZ)` on its cosets.\n\n EXAMPLES:\n\n Construct a noncongruence subgroup of index 7 (the smallest poss...

def HsuExample10(): '\n An example of an index 10 arithmetic subgroup studied by Tim Hsu.\n\n EXAMPLES::\n\n sage: import sage.modular.arithgroup.arithgroup_perm as ap\n sage: ap.HsuExample10()\n Arithmetic subgroup with permutations of right cosets\n S2=(1,2)(3,4)(5,6)(7,8)(9,1...

def HsuExample18(): '\n An example of an index 18 arithmetic subgroup studied by Tim Hsu.\n\n EXAMPLES::\n\n sage: import sage.modular.arithgroup.arithgroup_perm as ap\n sage: ap.HsuExample18()\n Arithmetic subgroup with permutations of right cosets\n S2=(1,5)(2,11)(3,10)(4,15)(...

def Gamma_constructor(N): '\n Return the congruence subgroup `\\Gamma(N)`.\n\n EXAMPLES::\n\n sage: Gamma(5) # indirect doctest\n Congruence Subgroup Gamma(5)\n sage: G = Gamma(23)\n sage: G is Gamma(23)\n True\n sage: TestSuite(G).run()\n\n Test global uniquenes...

@richcmp_method class Gamma_class(CongruenceSubgroup): '\n The principal congruence subgroup `\\Gamma(N)`.\n ' def _repr_(self): "\n Return the string representation of self.\n\n EXAMPLES::\n\n sage: Gamma(133)._repr_()\n 'Congruence Subgroup Gamma(133)'\n ...

def is_Gamma(x): '\n Return True if x is a congruence subgroup of type Gamma.\n\n EXAMPLES::\n\n sage: from sage.modular.arithgroup.all import is_Gamma\n sage: is_Gamma(Gamma0(13))\n False\n sage: is_Gamma(Gamma(4))\n True\n ' return isinstance(x, Gamma_class)

def _lift_pair(U, V, N): "\n Utility function. Given integers ``U, V, N``, with `N \\ge 1` and `{\\rm\n gcd}(U, V, N) = 1`, return a pair `(u, v)` congruent to `(U, V) \\bmod N`,\n such that `{\\rm gcd}(u,v) = 1`, `u, v \\ge 0`, `v` is as small as possible,\n and `u` is as small as possible for that `...

def is_Gamma0(x): '\n Return True if x is a congruence subgroup of type Gamma0.\n\n EXAMPLES::\n\n sage: from sage.modular.arithgroup.all import is_Gamma0\n sage: is_Gamma0(SL2Z)\n True\n sage: is_Gamma0(Gamma0(13))\n True\n sage: is_Gamma0(Gamma1(6))\n False...

def Gamma0_constructor(N): '\n Return the congruence subgroup Gamma0(N).\n\n EXAMPLES::\n\n sage: G = Gamma0(51) ; G # indirect doctest\n Congruence Subgroup Gamma0(51)\n sage: G == Gamma0(51)\n True\n sage: G is Gamma0(51)\n True\n ' from .all import SL2Z ...

class Gamma0_class(GammaH_class): "\n The congruence subgroup `\\Gamma_0(N)`.\n\n TESTS::\n\n sage: Gamma0(11).dimension_cusp_forms(2)\n 1\n sage: a = Gamma0(1).dimension_cusp_forms(2); a\n 0\n sage: type(a)\n <class 'sage.rings.integer.Integer'>\n sage: Gamm...

def is_Gamma1(x): '\n Return True if x is a congruence subgroup of type Gamma1.\n\n EXAMPLES::\n\n sage: from sage.modular.arithgroup.all import is_Gamma1\n sage: is_Gamma1(SL2Z)\n False\n sage: is_Gamma1(Gamma1(13))\n True\n sage: is_Gamma1(Gamma0(6))\n Fals...

def Gamma1_constructor(N): '\n Return the congruence subgroup `\\Gamma_1(N)`.\n\n EXAMPLES::\n\n sage: Gamma1(5) # indirect doctest\n Congruence Subgroup Gamma1(5)\n sage: G = Gamma1(23)\n sage: G is Gamma1(23)\n True\n sage: G is GammaH(23, [1])\n True\n ...

class Gamma1_class(GammaH_class): '\n The congruence subgroup `\\Gamma_1(N)`.\n\n TESTS::\n\n sage: [Gamma1(n).genus() for n in prime_range(2,100)]\n [0, 0, 0, 0, 1, 2, 5, 7, 12, 22, 26, 40, 51, 57, 70, 92, 117, 126, 155, 176, 187, 222, 247, 287, 345]\n sage: [Gamma1(n).index() for n in...

def GammaH_constructor(level, H): '\n Return the congruence subgroup `\\Gamma_H(N)`, which is the subgroup of\n `SL_2(\\ZZ)` consisting of matrices of the form `\\begin{pmatrix} a & b \\\\\n c & d \\end{pmatrix}` with `N | c` and `a, d \\in H`, for `H` a specified\n subgroup of `(\\ZZ/N\\ZZ)^\\times`....

def is_GammaH(x): '\n Return True if x is a congruence subgroup of type GammaH.\n\n EXAMPLES::\n\n sage: from sage.modular.arithgroup.all import is_GammaH\n sage: is_GammaH(GammaH(13, [2]))\n True\n sage: is_GammaH(Gamma0(6))\n True\n sage: is_GammaH(Gamma1(6))\n ...

def _normalize_H(H, level): '\n Normalize representatives for a given subgroup H of the units\n modulo level.\n\n .. NOTE::\n\n This function does *not* make any attempt to find a minimal\n set of generators for H. It simply normalizes the inputs for use\n in hashing.\n\n EXAMPLES...

@richcmp_method class GammaH_class(CongruenceSubgroup): '\n The congruence subgroup `\\Gamma_H(N)` for some subgroup `H \\trianglelefteq\n (\\ZZ / N\\ZZ)^\\times`, which is the subgroup of `\\SL_2(\\ZZ)` consisting of\n matrices of the form `\\begin{pmatrix} a &\n b \\\\ c & d \\end{pmatrix}` with `N ...

def _list_subgroup(N, gens): '\n Given an integer ``N`` and a list of integers ``gens``, return a list of\n the elements of the subgroup of `(\\ZZ / N\\ZZ)^\\times` generated by the\n elements of ``gens``.\n\n EXAMPLES::\n\n sage: sage.modular.arithgroup.congroup_gammaH._list_subgroup(11, [3])\...

def _GammaH_coset_helper(N, H): '\n Return a list of coset representatives for H in (Z / NZ)^*.\n\n EXAMPLES::\n\n sage: from sage.modular.arithgroup.congroup_gammaH import _GammaH_coset_helper\n sage: _GammaH_coset_helper(108, [1, 107])\n [1, 5, 7, 11, 13, 17, 19, 23, 25, 29, 31, 35, 3...

def mumu(N): '\n Return 0 if any cube divides `N`. Otherwise return\n `(-2)^v` where `v` is the number of primes that\n exactly divide `N`.\n\n This is similar to the Möbius function.\n\n INPUT:\n\n - ``N`` -- an integer at least 1\n\n OUTPUT: Integer\n\n EXAMPLES::\n\n sage: from s...

def CongruenceSubgroup_constructor(*args): '\n Attempt to create a congruence subgroup from the given data.\n\n The allowed inputs are as follows:\n\n - A :class:`~sage.groups.matrix_gps.matrix_group.MatrixGroup` object. This\n must be a group of matrices over `\\ZZ / N\\ZZ` for some `N`, with\n ...

def is_CongruenceSubgroup(x): '\n Return True if x is of type CongruenceSubgroup.\n\n Note that this may be False even if `x` really is a congruence subgroup --\n it tests whether `x` is "obviously" congruence, i.e.~whether it has a\n congruence subgroup datatype. To test whether or not an arithmetic ...

class CongruenceSubgroupBase(ArithmeticSubgroup): def __init__(self, level): '\n Create a congruence subgroup with given level.\n\n EXAMPLES::\n\n sage: Gamma0(500)\n Congruence Subgroup Gamma0(500)\n ' level = ZZ(level) if (level <= 0): ...

class CongruenceSubgroupFromGroup(CongruenceSubgroupBase): '\n A congruence subgroup, defined by the data of an integer `N` and a subgroup\n `G` of the finite group `SL(2, \\ZZ / N\\ZZ)`; the congruence subgroup\n consists of all the matrices in `SL(2, \\ZZ)` whose reduction modulo `N`\n lies in `G`.\...

class CongruenceSubgroup(CongruenceSubgroupFromGroup): '\n One of the "standard" congruence subgroups `\\Gamma_0(N)`, `\\Gamma_1(N)`,\n `\\Gamma(N)`, or `\\Gamma_H(N)` (for some `H`).\n\n This class is not intended to be instantiated directly. Derived subclasses\n must override ``_contains_sl2``, ``_r...

def _minimize_level(G): '\n Utility function. Given a matrix group `G` contained in `SL(2, \\ZZ / N\\ZZ)`\n for some `N`, test whether or not `G` is the preimage of a subgroup of\n smaller level, and if so, return that subgroup.\n\n The trivial group is handled specially: instead of returning a group,...

def is_SL2Z(x): '\n Return True if x is the modular group `\\SL_2(\\ZZ)`.\n\n EXAMPLES::\n\n sage: from sage.modular.arithgroup.all import is_SL2Z\n sage: is_SL2Z(SL2Z)\n True\n sage: is_SL2Z(Gamma0(6))\n False\n ' return isinstance(x, SL2Z_class)

class SL2Z_class(Gamma0_class): '\n The full modular group `\\SL_2(\\ZZ)`, regarded as a congruence\n subgroup of itself.\n ' def __init__(self): '\n The modular group `\\SL_2(\\Z)`.\n\n EXAMPLES::\n\n sage: G = SL2Z; G\n Modular Group SL(2,Z)\n ...

def _SL2Z_ref(): '\n Return SL2Z. (Used for pickling SL2Z.)\n\n EXAMPLES::\n\n sage: sage.modular.arithgroup.congroup_sl2z._SL2Z_ref()\n Modular Group SL(2,Z)\n sage: sage.modular.arithgroup.congroup_sl2z._SL2Z_ref() is SL2Z\n True\n ' return SL2Z

def random_even_arithgroup(index, nu2_max=None, nu3_max=None): '\n Return a random even arithmetic subgroup.\n\n EXAMPLES::\n\n sage: import sage.modular.arithgroup.tests as tests\n sage: G = tests.random_even_arithgroup(30); G # random\n Arithmetic subgroup of index 30\n sage: G...

def random_odd_arithgroup(index, nu3_max=None): '\n Return a random odd arithmetic subgroup.\n\n EXAMPLES::\n\n sage: from sage.modular.arithgroup.tests import random_odd_arithgroup\n sage: G = random_odd_arithgroup(20); G #random\n Arithmetic subgroup of index 20\n sage: G.is_od...

class Test(): '\n Testing class for arithmetic subgroup implemented via permutations.\n ' def __init__(self, index=20, index_max=50, odd_probability=0.5): '\n Create an arithmetic subgroup testing object.\n\n INPUT:\n\n - ``index`` - the index of random subgroup to test\n\n...

class DoubleCosetReduction(SageObject): "\n Edges in the Bruhat-Tits tree are represented by cosets of\n matrices in `GL_2`. Given a matrix `x` in `GL_2`, this\n class computes and stores the data corresponding to the\n double coset representation of `x` in terms of a fundamental\n domain of edges ...

class BruhatTitsTree(SageObject, UniqueRepresentation): '\n An implementation of the Bruhat-Tits tree for `GL_2(\\QQ_p)`.\n\n INPUT:\n\n - ``p`` - a prime number. The corresponding tree is then `p+1` regular\n\n EXAMPLES:\n\n We create the tree for `GL_2(\\QQ_5)`::\n\n sage: from sage.modula...

class Vertex(SageObject): '\n This is a structure to represent vertices of quotients of the\n Bruhat-Tits tree. It is useful to enrich the representation of\n the vertex as a matrix with extra data.\n\n INPUT:\n\n - ``p`` - a prime integer.\n\n - ``label`` - An integer which uniquely identifies...

class Edge(SageObject): '\n This is a structure to represent edges of quotients of the\n Bruhat-Tits tree. It is useful to enrich the representation of an\n edge as a matrix with extra data.\n\n INPUT:\n\n - ``p`` - a prime integer.\n\n - ``label`` - An integer which uniquely identifies this edg...

class BruhatTitsQuotient(SageObject, UniqueRepresentation): "\n This function computes the quotient of the Bruhat-Tits tree\n by an arithmetic quaternionic group. The group in question is the\n group of norm 1 elements in an Eichler `\\ZZ[1/p]`-order of some (tame)\n level inside of a definite quatern...

class _btquot_adjuster(Sigma0ActionAdjuster): '\n Callable object that turns matrices into 4-tuples.\n\n Since the modular symbol and harmonic cocycle code use different\n conventions for group actions, this function is used to make sure\n that actions are correct for harmonic cocycle computations.\n\...

def eval_dist_at_powseries(phi, f): '\n Evaluate a distribution on a powerseries.\n\n A distribution is an element in the dual of the Tate ring. The\n elements of coefficient modules of overconvergent modular symbols\n and overconvergent `p`-adic automorphic forms give examples of\n distributions i...

class BruhatTitsHarmonicCocycleElement(HeckeModuleElement): '\n `\\Gamma`-invariant harmonic cocycles on the Bruhat-Tits\n tree. `\\Gamma`-invariance is necessary so that the cocycle can be\n stored in terms of a finite amount of data.\n\n More precisely, given a ``BruhatTitsQuotient`` `T`, harmonic c...

class BruhatTitsHarmonicCocycles(AmbientHeckeModule, UniqueRepresentation): '\n Ensure unique representation\n\n EXAMPLES::\n\n sage: X = BruhatTitsQuotient(3,5)\n sage: M1 = X.harmonic_cocycles( 2, prec = 10)\n sage: M2 = X.harmonic_cocycles( 2, 10)\n sage: M1 is M2\n Tru...

class pAdicAutomorphicFormElement(ModuleElement): "\n Rudimentary implementation of a class for a `p`-adic\n automorphic form on a definite quaternion algebra over `\\QQ`. These\n are required in order to compute moments of measures associated to\n harmonic cocycles on the Bruhat-Tits tree using the o...

class pAdicAutomorphicForms(Module, UniqueRepresentation): Element = pAdicAutomorphicFormElement @staticmethod def __classcall__(cls, domain, U, prec=None, t=None, R=None, overconvergent=False): '\n The module of (quaternionic) `p`-adic automorphic forms.\n\n INPUT:\n\n - ``d...

def gp(): '\n Return a copy of the GP interpreter with the appropriate files loaded.\n\n EXAMPLES::\n\n sage: import sage.modular.buzzard\n sage: sage.modular.buzzard.gp()\n PARI/GP interpreter\n ' global _gp if (_gp is None): _gp = Gp(script_subdirectory='buzzard') ...

def buzzard_tpslopes(p, N, kmax): "\n Return a vector of length kmax, whose `k`'th entry\n (`0 \\leq k \\leq k_{max}`) is the conjectural sequence\n of valuations of eigenvalues of `T_p` on forms of level\n `N`, weight `k`, and trivial character.\n\n This conjecture is due to Kevin Buzzard, and is ...

class Cusp(Element): '\n A cusp.\n\n A cusp is either a rational number or infinity, i.e., an element of\n the projective line over Q. A Cusp is stored as a pair (a,b), where\n gcd(a,b)=1 and a,b are of type Integer.\n\n EXAMPLES::\n\n sage: a = Cusp(2/3); b = Cusp(oo)\n sage: a.paren...

class Cusps_class(Singleton, Parent): '\n The set of cusps.\n\n EXAMPLES::\n\n sage: C = Cusps; C\n Set P^1(QQ) of all cusps\n sage: loads(C.dumps()) == C\n True\n ' def __init__(self): '\n The set of cusps, i.e. `\\mathbb{P}^1(\\QQ)`.\n\n EXAMPLES::...

@cached_function def list_of_representatives(N): "\n Return a list of ideals, coprime to the ideal ``N``, representatives of\n the ideal classes of the corresponding number field.\n\n .. NOTE::\n\n This list, used every time we check `\\Gamma_0(N)` - equivalence of\n cusps, is cached.\n\n ...

@cached_function def NFCusps(number_field): "\n The set of cusps of a number field `K`, i.e. `\\mathbb{P}^1(K)`.\n\n INPUT:\n\n - ``number_field`` -- a number field\n\n OUTPUT:\n\n The set of cusps over the given number field.\n\n EXAMPLES::\n\n sage: x = polygen(ZZ, 'x')\n sage: k...

class NFCuspsSpace(UniqueRepresentation, Parent): "\n The set of cusps of a number field. See ``NFCusps`` for full documentation.\n\n EXAMPLES::\n\n sage: x = polygen(ZZ, 'x')\n sage: k.<a> = NumberField(x^2 + 5)\n sage: kCusps = NFCusps(k); kCusps\n Set of all cusps of Number Fi...

class NFCusp(Element): '\n Create a number field cusp, i.e., an element of `\\mathbb{P}^1(k)`.\n\n A cusp on a number field is either an element of the field or infinity,\n i.e., an element of the projective line over the number field. It is\n stored as a pair (a,b), where a, b are integral elements ...

def Gamma0_NFCusps(N): "\n Return a list of inequivalent cusps for `\\Gamma_0(N)`, i.e., a set of\n representatives for the orbits of ``self`` on `\\mathbb{P}^1(k)`.\n\n INPUT:\n\n - ``N`` -- an integral ideal of the number field k (the level).\n\n OUTPUT:\n\n A list of inequivalent number field...

def number_of_Gamma0_NFCusps(N): "\n Return the total number of orbits of cusps under the action of the\n congruence subgroup `\\Gamma_0(N)`.\n\n INPUT:\n\n - ``N`` -- a number field ideal.\n\n OUTPUT:\n\n integer -- the number of orbits of cusps under Gamma0(N)-action.\n\n EXAMPLES::\n\n ...

def NFCusps_ideal_reps_for_levelN(N, nlists=1): "\n Return a list of lists (``nlists`` different lists) of prime ideals,\n coprime to ``N``, representing every ideal class of the number field.\n\n INPUT:\n\n - ``N`` -- number field ideal.\n\n - ``nlists`` -- optional (default 1). The number of list...

def units_mod_ideal(I): "\n Return integral elements of the number field representing the images of\n the global units modulo the ideal ``I``.\n\n INPUT:\n\n - ``I`` -- number field ideal.\n\n OUTPUT:\n\n A list of integral elements of the number field representing the images of\n the global ...

def eisen(p): '\n Return the Eisenstein number `n` which is the numerator of `(p-1)/12`.\n\n INPUT:\n\n - ``p`` -- a prime\n\n OUTPUT: Integer\n\n EXAMPLES::\n\n sage: [(p, sage.modular.dims.eisen(p)) for p in prime_range(24)]\n [(2, 1), (3, 1), (5, 1), (7, 1), (11, 5), (13, 1), (17, ...

def CO_delta(r, p, N, eps): '\n This is used as an intermediate value in computations related to\n the paper of Cohen-Oesterlé.\n\n INPUT:\n\n - ``r`` -- positive integer\n\n - ``p`` -- a prime\n\n - ``N`` -- positive integer\n\n - ``eps`` -- character\n\n OUTPUT: element of the base r...

def CO_nu(r, p, N, eps): '\n This is used as an intermediate value in computations related to\n the paper of Cohen-Oesterlé.\n\n INPUT:\n\n - ``r`` -- positive integer\n\n - ``p`` -- a prime\n\n - ``N`` -- positive integer\n\n - ``eps`` -- character\n\n OUTPUT: element of the base ring...

def CohenOesterle(eps, k): '\n Compute the Cohen-Oesterlé function associate to eps, `k`.\n\n This is a summand in the formula for the dimension of the space of\n cusp forms of weight `2` with character `\\varepsilon`.\n\n INPUT:\n\n - ``eps`` -- Dirichlet character\n\n - ``k`` -- integer\n\n ...

def dimension_new_cusp_forms(X, k=2, p=0): '\n Return the dimension of the new (or `p`-new) subspace of\n cusp forms for the character or group `X`.\n\n INPUT:\n\n - ``X`` -- integer, congruence subgroup or Dirichlet\n character\n\n - ``k`` -- weight (integer)\n\n - ``p`` -- 0 or a prim...

def dimension_cusp_forms(X, k=2): '\n The dimension of the space of cusp forms for the given congruence\n subgroup or Dirichlet character.\n\n INPUT:\n\n - ``X`` -- congruence subgroup or Dirichlet character\n or integer\n\n - ``k`` -- weight (integer)\n\n EXAMPLES::\n\n sage: fro...

def dimension_eis(X, k=2): '\n The dimension of the space of Eisenstein series for the given\n congruence subgroup.\n\n INPUT:\n\n - ``X`` -- congruence subgroup or Dirichlet character\n or integer\n\n - ``k`` -- weight (integer)\n\n EXAMPLES::\n\n sage: from sage.modular.dims imp...

def dimension_modular_forms(X, k=2): '\n The dimension of the space of cusp forms for the given congruence\n subgroup (either `\\Gamma_0(N)`, `\\Gamma_1(N)`, or\n `\\Gamma_H(N)`) or Dirichlet character.\n\n INPUT:\n\n - ``X`` -- congruence subgroup or Dirichlet character\n\n - ``k`` -- weight ...

def sturm_bound(level, weight=2): '\n Return the Sturm bound for modular forms with given level and weight.\n\n For more details, see the documentation for the ``sturm_bound`` method\n of :class:`sage.modular.arithgroup.CongruenceSubgroup` objects.\n\n INPUT:\n\n - ``level`` -- an integer (interpre...

def trivial_character(N, base_ring=QQ): '\n Return the trivial character of the given modulus, with values in the given\n base ring.\n\n EXAMPLES::\n\n sage: t = trivial_character(7)\n sage: [t(x) for x in [0..20]]\n [0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1]\n ...

def kronecker_character(d): '\n Return the quadratic Dirichlet character (d/.) of minimal\n conductor.\n\n EXAMPLES::\n\n sage: kronecker_character(97*389*997^2)\n Dirichlet character modulo 37733 of conductor 37733 mapping 1557 |--> -1, 37346 |--> -1\n\n ::\n\n sage: a = kronecke...

def kronecker_character_upside_down(d): '\n Return the quadratic Dirichlet character (./d) of conductor d, for\n d > 0.\n\n EXAMPLES::\n\n sage: kronecker_character_upside_down(97*389*997^2)\n Dirichlet character modulo 37506941597 of conductor 37733 mapping 13533432536 |--> -1, 22369178537...

def is_DirichletCharacter(x) -> bool: '\n Return ``True`` if ``x`` is of type ``DirichletCharacter``.\n\n EXAMPLES::\n\n sage: from sage.modular.dirichlet import is_DirichletCharacter\n sage: is_DirichletCharacter(trivial_character(3))\n True\n sage: is_DirichletCharacter([1])\n ...

class DirichletCharacter(MultiplicativeGroupElement): '\n A Dirichlet character.\n ' def __init__(self, parent, x, check=True): '\n Create a Dirichlet character with specified values on\n generators of `(\\ZZ/n\\ZZ)^*`.\n\n INPUT:\n\n - ``parent`` -- :class:`Dirichle...

class DirichletGroupFactory(UniqueFactory): '\n Construct a group of Dirichlet characters modulo `N`.\n\n INPUT:\n\n - ``N`` -- positive integer\n\n - ``base_ring`` -- commutative ring; the value ring for the\n characters in this group (default: the cyclotomic field\n `\\QQ(\\zeta_n)`, where...

def is_DirichletGroup(x): '\n Return ``True`` if ``x`` is a Dirichlet group.\n\n EXAMPLES::\n\n sage: from sage.modular.dirichlet import is_DirichletGroup\n sage: is_DirichletGroup(DirichletGroup(11))\n True\n sage: is_DirichletGroup(11)\n False\n sage: is_Dirichlet...

class DirichletGroup_class(WithEqualityById, Parent): '\n Group of Dirichlet characters modulo `N` with values in a ring `R`.\n ' Element = DirichletCharacter def __init__(self, base_ring, modulus, zeta, zeta_order): '\n Create a Dirichlet group.\n\n Not to be called directly ...

def EtaGroup(level): '\n Create the group of eta products of the given level.\n\n EXAMPLES::\n\n sage: EtaGroup(12)\n Group of eta products on X_0(12)\n sage: EtaGroup(1/2)\n Traceback (most recent call last):\n ...\n TypeError: Level (=1/2) must be a positive integ...

class EtaGroupElement(Element): def __init__(self, parent, rdict): '\n Create an eta product object. Usually called implicitly via\n EtaGroup_class.__call__ or the EtaProduct factory function.\n\n EXAMPLES::\n\n sage: EtaProduct(8, {1:24, 2:-24})\n Eta product o...

class EtaGroup_class(UniqueRepresentation, Parent): '\n The group of eta products of a given level under multiplication.\n\n TESTS::\n\n sage: TestSuite(EtaGroup(12)).run()\n\n sage: EtaGroup(12) == EtaGroup(12)\n True\n sage: EtaGroup(12) == EtaGroup(13)\n False\n\n ...

def EtaProduct(level, dic) -> EtaGroupElement: "\n Create an :class:`EtaGroupElement` object representing the function\n `\\prod_{d | N} \\eta(q^d)^{r_d}`.\n\n This checks the criteria of Ligozat to ensure that this product\n really is the `q`-expansion of a meromorphic function on `X_0(N)`.\n\n IN...

def num_cusps_of_width(N, d) -> Integer: '\n Return the number of cusps on `X_0(N)` of width ``d``.\n\n INPUT:\n\n - ``N`` -- (integer): the level\n\n - ``d`` -- (integer): an integer dividing N, the cusp width\n\n EXAMPLES::\n\n sage: from sage.modular.etaproducts import num_cusps_of_widt...

def AllCusps(N): '\n Return a list of CuspFamily objects corresponding to the cusps of\n `X_0(N)`.\n\n INPUT:\n\n - ``N`` -- (integer): the level\n\n EXAMPLES::\n\n sage: AllCusps(18)\n [(Inf), (c_{2}), (c_{3,1}), (c_{3,2}), (c_{6,1}), (c_{6,2}), (c_{9}), (0)]\n sage: AllCusps...

@richcmp_method class CuspFamily(SageObject): '\n A family of elliptic curves parametrising a region of `X_0(N)`.\n ' def __init__(self, N, width, label=None): "\n Create the cusp of width d on X_0(N) corresponding to the family\n of Tate curves `(\\CC_p/q^d, \\langle \\zeta q\\ra...

def qexp_eta(ps_ring, prec): "\n Return the q-expansion of `\\eta(q) / q^{1/24}`.\n\n Here `\\eta(q)` is Dedekind's function\n\n .. MATH::\n\n \\eta(q) = q^{1/24}\\prod_{n=1}^\\infty (1-q^n).\n\n The result is an element of ``ps_ring``, with precision ``prec``.\n\n INPUT:\n\n - ``ps_ring...

def eta_poly_relations(eta_elements, degree, labels=['x1', 'x2'], verbose=False): "\n Find polynomial relations between eta products.\n\n INPUT:\n\n - ``eta_elements`` - (list): a list of EtaGroupElement objects.\n Not implemented unless this list has precisely two elements. degree\n\n - ``degree...

def _eta_relations_helper(eta1, eta2, degree, qexp_terms, labels, verbose): "\n Helper function used by eta_poly_relations. Finds a basis for the\n space of linear relations between the first qexp_terms of the\n `q`-expansions of the monomials\n `\\eta_1^i * \\eta_2^j` for `0 \\le i,j < degree`,\n ...

def is_HeckeAlgebra(x): '\n Return ``True`` if x is of type HeckeAlgebra.\n\n EXAMPLES::\n\n sage: from sage.modular.hecke.algebra import is_HeckeAlgebra\n sage: is_HeckeAlgebra(CuspForms(1, 12).anemic_hecke_algebra())\n True\n sage: is_HeckeAlgebra(ZZ)\n False\n ' ...

def _heckebasis(M): '\n Return a basis of the Hecke algebra of M as a ZZ-module.\n\n INPUT:\n\n - ``M`` -- a Hecke module\n\n OUTPUT:\n\n a list of Hecke algebra elements represented as matrices\n\n EXAMPLES::\n\n sage: M = ModularSymbols(11,2,1)\n sage: sage.modular.hecke.algebra....

@richcmp_method class HeckeAlgebra_base(CachedRepresentation, CommutativeAlgebra): '\n Base class for algebras of Hecke operators on a fixed Hecke module.\n\n INPUT:\n\n - ``M`` - a Hecke module\n\n EXAMPLES::\n\n sage: CuspForms(1, 12).hecke_algebra() # indirect doctest\n Full Hecke al...

class HeckeAlgebra_full(HeckeAlgebra_base): '\n A full Hecke algebra (including the operators `T_n` where `n` is not\n assumed to be coprime to the level).\n ' def _repr_(self): "\n String representation of self.\n\n EXAMPLES::\n\n sage: ModularForms(37).hecke_algebr...

class HeckeAlgebra_anemic(HeckeAlgebra_base): '\n An anemic Hecke algebra, generated by Hecke operators with index coprime to the level.\n ' def _repr_(self): '\n EXAMPLES::\n\n sage: H = CuspForms(3, 12).anemic_hecke_algebra()._repr_()\n ' return ('Anemic Hecke...

def is_AmbientHeckeModule(x) -> bool: '\n Return ``True`` if ``x`` is of type ``AmbientHeckeModule``.\n\n EXAMPLES::\n\n sage: from sage.modular.hecke.ambient_module import is_AmbientHeckeModule\n sage: is_AmbientHeckeModule(ModularSymbols(6))\n True\n sage: is_AmbientHeckeModule...

class AmbientHeckeModule(module.HeckeModule_free_module): '\n An ambient Hecke module, i.e. a Hecke module that is isomorphic as a module\n over its base ring `R` to the standard free module `R^k` for some `k`. This\n is the base class for ambient spaces of modular forms and modular symbols,\n and for...

class DegeneracyMap(morphism.HeckeModuleMorphism_matrix): '\n A degeneracy map between Hecke modules of different levels.\n\n EXAMPLES:\n\n We construct a number of degeneracy maps::\n\n sage: M = ModularSymbols(33)\n sage: d = M.degeneracy_map(11)\n sage: d\n Hecke module mor...

def is_HeckeModuleElement(x): '\n Return ``True`` if x is a Hecke module element, i.e., of type HeckeModuleElement.\n\n EXAMPLES::\n\n sage: sage.modular.hecke.all.is_HeckeModuleElement(0)\n False\n sage: sage.modular.hecke.all.is_HeckeModuleElement(BrandtModule(37)([1,2,3]))\n T...

class HeckeModuleElement(ModuleElement): '\n Element of a Hecke module.\n ' def __init__(self, parent, x=None): "\n INPUT:\n\n - ``parent`` -- a Hecke module\n\n - ``x`` -- element of the free module associated to parent\n\n EXAMPLES::\n\n sage: v = sage.m...

def is_HeckeOperator(x): '\n Return ``True`` if x is of type HeckeOperator.\n\n EXAMPLES::\n\n sage: from sage.modular.hecke.hecke_operator import is_HeckeOperator\n sage: M = ModularSymbols(Gamma0(7), 4)\n sage: is_HeckeOperator(M.T(3))\n True\n sage: is_HeckeOperator(M.T...

def is_HeckeAlgebraElement(x): '\n Return ``True`` if x is of type HeckeAlgebraElement.\n\n EXAMPLES::\n\n sage: from sage.modular.hecke.hecke_operator import is_HeckeAlgebraElement\n sage: M = ModularSymbols(Gamma0(7), 4)\n sage: is_HeckeAlgebraElement(M.T(3))\n True\n sa...

class HeckeAlgebraElement(AlgebraElement): '\n Base class for elements of Hecke algebras.\n ' def __init__(self, parent): "\n Create an element of a Hecke algebra.\n\n EXAMPLES::\n\n sage: R = ModularForms(Gamma0(7), 4).hecke_algebra()\n sage: sage.modular.he...

class HeckeAlgebraElement_matrix(HeckeAlgebraElement): '\n An element of the Hecke algebra represented by a matrix.\n ' def __init__(self, parent, A): "\n Initialise an element from a matrix. This *must* be over the base ring\n of self and have the right size.\n\n This is a...

class DiamondBracketOperator(HeckeAlgebraElement_matrix): '\n The diamond bracket operator `\\langle d \\rangle` for some `d \\in \\ZZ /\n N\\ZZ` (which need not be a unit, although if it is not, the operator will\n be zero).\n ' def __init__(self, parent, d): "\n Standard init fun...

class HeckeOperator(HeckeAlgebraElement): '\n The Hecke operator `T_n` for some `n` (which need not be coprime to the\n level). The matrix is not computed until it is needed.\n ' def __init__(self, parent, n): '\n EXAMPLES::\n\n sage: M = ModularSymbols(11)\n sag...