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 A mf.gl2.history A brief history of $\textrm{GL}(2)$ modular forms curve.highergenus.aut.group_action A group acting on a compact Riemann Surface lfunction.history A history of L-functions intro.tutorial A tour of the Riemann zeta function group.a_group A-group belyi.abc abc triple for Belyi maps group.abelian Abelian group nf.abelian Abelian number field ag.abelian_surface Abelian surface av.fq.one_rational_point Abelian varieties with one rational point ag.abelian_variety Abelian variety group.abelianization Abelianization group.abelianization_isolabel Abelianization label varieties.about About varieties g2c.abs_discriminant Absolute discriminant nf.abs_discriminant Absolute discriminant of a number field group.galois.absolute Absolute Galois group nf.absolute_value Absolute value of a field group.name Abstract group name rcs.ack.groups.abstract Acknowledgements for abstract groups rcs.ack.modcurve Acknowledgements for modular curve data rcs.ack.lfunction.curve Acknowledgments for (non-elliptic) curve L-functions rcs.ack.artin Acknowledgments for Artin representations rcs.ack.belyi Acknowledgments for Belyi maps rcs.ack.mf.bianchi Acknowledgments for Bianchi modular forms rcs.ack.cmf Acknowledgments for classical modular forms rcs.ack.curve.highergenus.aut Acknowledgments for curves of higher genus with automorphism group data rcs.ack.character.dirichlet Acknowledgments for Dirichlet characters rcs.ack.lfunction.ec Acknowledgments for elliptic curve L-functions rcs.ack.ec.q Acknowledgments for elliptic curves over $\Q$ rcs.ack.gg Acknowledgments for Galois groups rcs.ack.g2c Acknowledgments for genus 2 curves rcs.ack.mf.hilbert Acknowledgments for Hilbert modular forms rcs.ack.hosting Acknowledgments for hosting rcs.ack.lfunction Acknowledgments for L-functions rcs.ack.lfunction.maass Acknowledgments for Maass form L-functions rcs.ack.maass Acknowledgments for Maass forms rcs.ack.meetings Acknowledgments for meetings rcs.ack.lfunction.modular Acknowledgments for modular form L-functions rcs.ack.nf Acknowledgments for number fields rcs.ack.lf Acknowledgments for p-adic fields rcs.ack.st_group Acknowledgments for Sato-Tate groups rcs.ack.software Acknowledgments for software rcs.ack.sponsors Acknowledgments for sponsors group.fuchsian.action Action of $\SL(2,\R)$ in the upper half-plane rcs.ack.general Additional acknowledgments ec.additive_reduction Additive reduction ag.affine_space Affine space nf.algebraic_closure Algebraic closure of a field ag.curve Algebraic curve mf.gl2.history.varieties Algebraic varieties ag.variety Algebraic variety group.almost_simple Almost simple group group.alternating Alternating group group.ambient Ambient group group.ambient_isolabel Ambient group label st_group.ambient Ambient group of a Sato-Tate group nf.class_number_formula Analytic class number formula lfunction.analytic_conductor Analytic conductor cmf.analytic_conductor Analytic conductor of a classical newform ec.analytic_sha_order Analytic order of Ш ec.q.analytic_sha_order Analytic order of Ш g2c.analytic_sha Analytic order of Sha ec.q.analytic_sha_value Analytic order of Sha (numerical value) cmf.analytic_rank Analytic rank lfunction.analytic_rank Analytic rank modcurve.rank Analytic rank mf.bianchi.anr Analytic rank of a Bianchi newform g2c.analytic_rank Analytic rank of a curve ec.q.analytic_rank Analytic rank of an elliptic curve over $\Q$ av.fq.angle_rank Angle rank of a set lfunction.arithmetic Arithmetic L-function nf.arithmetically_equivalent Arithmetically equivalent fields gg.arithmetically_equivalent Arithmetically equivalent siblings cmf.artin_field Artin field cmf.artin_image Artin image artin.lfunction Artin L-function lfunction.history.artin Artin L-functions artin.stem_field Artin stem field ring.associate Associate of an element alg.binary_operation.associative Associative binary operation nf.assuming_grh Assuming GRH cmf.atkin-lehner Atkin-Lehner involution $w_Q$ group.autjugacy_class Autjugacy class group.autjugate_subgroup Autjugate subgroups mf.siegel.automorphic_type Automorphic type curve.highergenus.aut.characters Automorphism group characters lf.automorphism_group Automorphism group of a field extension g2c.aut_grp Automorphism group of an algebraic curve lattice.automorphism_group Automorphism group of an integral lattice group.automorphism Automorphisms of a group mf.siegel.extent.kp Available samples for $M_2(K(p))$ mf.siegel.extent.sp4z Available samples for $M_k(\mathrm{Sp}(4,\mathbb{Z}))$ mf.siegel.extent.sp6z Available samples for $M_k(\mathrm{Sp}(6,\mathbb{Z}))$ mf.siegel.extent.sp8z Available samples for $M_k(\mathrm{Sp}(8,\mathbb{Z}))$ mf.siegel.extent.sp4z_2 Available samples for $M_{k,2}(\mathrm{Sp}(4,\mathbb{Z}))$ B cmf.bad_prime Bad prime lfunction.bad_prime Bad prime ec.bad_reduction Bad reduction of an elliptic curve at a prime ag.base_change Base change ec.base_change Base change mf.base_change Base change of a GL(2) modular form ag.base_field Base field belyi.base_field Base field of a Belyi map mf.bianchi.base_change Base-change Bianchi modular forms character.dirichlet.basic_properties Basic properties of Dirichlet characters belyi.galmap Belyi map belyi.label Belyi map label belyi.orders Belyi permutation triple orders af.bernoulli_numbers Bernoulli numbers mf.bianchi.bianchigroup Bianchi Groups mf.bianchi Bianchi modular form mf.bianchi.bianchimodularforms Bianchi modular forms mf.bianchi.newform Bianchi newform alg.binary_operation Binary operation ec.q.bsdconjecture Birch and Swinnerton-Dyer conjecture ec.q.bsd_invariants Birch and Swinnerton-Dyer invariants of an elliptic curve/$\mathbb{Q}$ ec.bsdconjecture Birch Swinnerton-Dyer conjecture gl2.borel Borel subgroup curve.highergenus.aut.braid_equivalence Braid Equivalence curve.highergenus.aut.branchpoints Branch Points g2c.bsd_invariants BSD invariants stats.buckets Buckets for statistics hgm.bezout_determinant Bézout determinant alg.bezout_matrix Bézout matrix hgm.bezout_matrix Bézout matrix of a hypergeometric motive C nf.polredabs Canonical defining polynomial for number fields modcurve.canonical_field Canonical field of definition of a modular curve ec.q.canonical_height Canonical height ag.canonical_height Canonical height of an abelian variety ec.canonical_height Canonical height on an elliptic curve gl2.cartan Cartan subgroup lattice.catalogue_of_lattices Catalogue of Lattices group.center_isolabel Center label group.center Center of a group lfunction.central_character Central character of an L-function lfunction.central_point Central point of an L-function group.central_quotient Central quotient group.central_quotient_isolabel Central quotient label group.central Central subgroup lfunction.central_value Central value of an L-function group.centralizer Centralizer of a subgroup group.subgroup.centralizer Centralizer of a subgroup group.representation.character Character of a group representation mf.maass.mwf.character Character of a Maass form cmf.character Character of a modular form gg.character_table.data Character table data ring.characteristic Characteristic of a ring av.fq.weil_polynomial Characteristic polynomial of an abelian variety over a finite field group.characteristic_subgroup Characteristic subgroup group.chief_series Chief series of a group nf.class_number Class number of a number field lattice.class_number Class number of an integral lattice ag.cluster_picture Cluster picture mf.bianchi.cm CM Bianchi newforms ag.cm_field CM field cmf.cm_form CM form nf.cm_field CM number field modcurve.cm_discriminants CM points cmf.minimal_twist cmf.minimal_twist cmf.coefficient_field Coefficient field for newforms mf.siegel.coefficient_field Coefficient field for Siegel newforms lfunction.coefficient_field Coefficient field of an L-function cmf.coefficient_ring Coefficient ring mf.siegel.coefficient_ring Coefficient ring cmf.hecke_ring_generators Coefficient ring generator bound mf.gl2.history.combinatorics Combinatorics and $q$-series alg.binary_operation.commutative Commutative binary operation group.commutator_isolabel Commutator label group.commutator_length Commutator length group.commutator_subgroup Commutator subgroup of a group group.subgroup.complement Complement group.complete Complete group lfunction.completed Completed L-function rcs.cande.groups.abstract Completeness and extent of abstract group data rcs.cande.artin Completeness and extent of Artin representation data rcs.cande.gg Completeness and extent of Galois group data rcs.cande.lf Completeness and extent of local field data rcs.cande.maass Completeness and extent of Maass form data dq.mf.bianchi.extent Completeness of Bianchi Modular Form data rcs.cande.mf.bianchi Completeness of Bianchi modular form data rcs.cande.cmf Completeness of Classical Modular Form Data dq.belyi.extent Completeness of data for Belyi maps rcs.cande.belyi Completeness of data for Belyi maps rcs.cande.curve.highergenus.aut Completeness of data for families of higher genus curves with automorphisms dq.lattice.completeness Completeness of data for Integral Lattices rcs.cande.lattice Completeness of data for Integral Lattices dq.hgm.extent Completeness of data on hypergeometric motives rcs.cande.hgm Completeness of data on hypergeometric motives rcs.cande.character.dirichlet Completeness of Dirichlet character data rcs.cande.ec Completeness of elliptic curve data rcs.cande.ec.q Completeness of elliptic curve data over $\Q$ rcs.cande.g2c Completeness of genus 2 curve data dq.mf.hilbert.extent Completeness of Hilbert modular form data rcs.cande.mf.hilbert Completeness of Hilbert modular form data rcs.cande.lfunction Completeness of L-function Data rcs.cande.zeros.zeta Completeness of Riemann Zeta Zeros Data rcs.cande.st_group Completeness of Sato-Tate Group Data st_group.summary Completeness of Sato-Tate group data rcs.cande.modcurve Completeness of the modular curve database group.complex_character_table Complex character table of a finite group nf.complex_embedding Complex embedding cmf.embedding_label Complex embedding label group.representation.irrep Complex irreducible representation ag.complex_multiplication Complex multiplication ec.complex_multiplication Complex multiplication st_group.component_group Component group of a Sato-Tate group st_group.components Components of a Sato-Tate group g2c.conditional_mw_group Conditional Mordell-Weil group ec.conductor_label Conductor labels for elliptic curves character.dirichlet.conductor Conductor of a Dirichlet character character.dirichlet.conrey.conductor Conductor of a Dirichlet character from its Conrey label hgm.conductor Conductor of a hypergeometric motive nf.conductor Conductor of an abelian number field ag.conductor Conductor of an abelian variety artin.conductor Conductor of an Artin representation ec.conductor Conductor of an elliptic curve ec.q.conductor Conductor of an elliptic curve over $\Q$ lfunction.conductor Conductor of an L-function ec.conductor_valuation Conductor valuation cmf.congruence_subgroup Congruence subgroup mf.bianchi.bianchicongruencesubgroup Congruence subgroup of a Bianchi group group.sp2gr.subgroup.congn Congruence subgroup of level $N$ of $\Sp(2g,\Z)$ ec.congruent_number_curve Congruent number curves ec.congruent_number_problem Congruent number problem ec.congruent_number Congruent numbers gg.conjugacy_classes Conjugacy class group.conjugacy_class Conjugacy class gg.conjugacy_classes.data Conjugacy class representatives curve.highergenus.aut.conjugacyclasses Conjugacy classes st_group.connected Connected Sato-Tate group character.dirichlet.conrey.index Conrey index of a Dirichlet character character.dirichlet.conrey Conrey label of Dirichlet characters character.dirichlet.conrey.primitive_generator Conrey primitive generator modulo $p$ artin.permutation_container Containing permutation representation modcurve.contains_negative_one Contains $-I$ group.core Core of a subgroup av.fq.frobenius_angles_correctness Correctness of Frobenius angles group.coset Coset of a subgroup doc.create_knowl Create a knowl ec.q.cremona_label Cremona label lfunction.critical_line Critical line of an L-function lfunction.critical_strip Critical strip crystals.crystal Crystal hgm.cusp_index Cusp index cmf.cusp_form Cuspidal modular form modcurve.cusps Cusps of a modular curve group.fuchsian.cusps Cusps of a subgroup of the modular group group.cyclic Cyclic group ag.cyclic_trigonal Cyclic trigonal curve D clusterpicture.data Data for a cluster picture character.dirichlet.data Data for a dirichlet character character.dirichlet.orbit_data Data for a Galois orbit of Dirichlet characters nf.field.data Data for a global number field lf.field.data Data for a local field lattice.data Data for an integral lattice group.small.data Data on small group av.decomposition Decomposition into isogeny factors cmf.decomposition.new.gamma0chi Decomposition into newforms cmf.dim_decomposition Decomposition into newforms cmf.decomposition.new.gamma1 Decomposition into newforms for Gamma1(N) g2c.decomposition Decomposition of the Jacobian of a genus 2 curve ring.dedekind_domain Dedekind domain mf.half_integral_weight.dedekind_eta Dedekind eta function lfunction.history.dedekind Dedekind zeta-functions, history hgm.defining_parameters Defining parameters of a hypergeometric motive cmf.defining_polynomial Defining polynomial mf.siegel.defining_polynomial Defining polynomial lf.defining_polynomial Defining polynomial of a local field nf.defining_polynomial Defining Polynomial of a Number Field lf.qp Definition of $\Q_p$ group Definition of group gg.degree Degree lfunction.known_degree1 Degree 1 L-functions, all lfunction.known_degree2 Degree 2 L-functions, examples lfunction.known_degree3 Degree 3 L-functions, examples lfunction.known_degree4 Degree 4 L-functions, examples belyi.degree Degree of a Belyi map character.dirichlet.degree Degree of a Dirichlet character hgm.degree Degree of a hypergeometric motive lf.degree Degree of a local field extension nf.degree Degree of a number field st_group.degree Degree of a Sato-Tate group mf.siegel.degree Degree of a Siegel modular form lfunction.degree Degree of an L-function lattice.density Density of an integral lattice group.derived_series Derived series of a group doc.knowl.description Description field in knowls cmf.statistics_extent Description of completeness artin.determinant Determinant of an Artin representation lattice.determinant Determinant of an integral lattice doc.development Developers Guide group.dicyclic Dicyclic group group.dihedral Dihedral group cmf.dimension Dimension cmf.display_dim Dimension mf.siegel.dimension Dimension curve.highergenus.aut.dimension Dimension of a family of curves cmf.dimension_galois_orbit Dimension of a Hecke orbit mf.hilbert.dimension Dimension of a Hecke orbit ag.dimension Dimension of an algebraic variety artin.dimension Dimension of an Artin representation lattice.dimension Dimension of an integral lattice cmf.atkin_lehner_dims Dimensions of Atkin-Lehner subspaces group.direct_product Direct product of groups character.dirichlet Dirichlet character artin.dimensionone Dirichlet character associated with a 1-dimensional Artin representation dq.character.dirichlet.extent Dirichlet character data extent character.dirichlet.group Dirichlet character group nf.dirichlet_group Dirichlet group of an Abelian number field lfunction.dirichlet Dirichlet L-function lfunction.dirichlet_series Dirichlet series lfunction.history.dirichlet Dirichlet's work on L-functions lf.discriminant_exponent Discriminant exponent of a local field g2c.discriminant Discriminant of a genus 2 curve nf.discriminant Discriminant of a number field ec.discriminant Discriminant of a Weierstrass equation ec.q.discriminant Discriminant of an elliptic curve over $\Q$ nf.poly_discriminant Discriminant of polynomial lf.discriminant_root_field Discriminant root field of a local field ec.discriminant_valuation Discriminant valuation cmf.distinguishing_primes Distinguishing Hecke operators group.division Divisions of a group af.divisor_function Divisor function cmf.dualform Dual cuspform lattice.dual Dual lattice lfunction.dual Dual of an L-function mf.siegel.lift.ikeda Duke-Imamoglu-Ikeda lift E lf.eisenstein_polynomial Eisenstein polynomial group.elementary Elementary group ec.invariants Elliptic curve invariants ec.q.invariants Elliptic curve invariants ec.ring Elliptic curve over a ring mf.gl2.history.elliptic Elliptic functions ec.scheme Elliptic scheme gg.elusive_group Elusive group cmf.embedding_format Embedding Format cmf.embedding Embedding of a modular form nf.embedding Embedding of a number field st_group.embedding Embedding of a Sato-Tate group ag.endomorphism_algebra Endomorphism algebra g2c.end_alg Endomorphism algebra av.endomorphism_field Endomorphism field g2c.jac_endomorphisms Endomorphism invariants g2c.jac_end_lattice Endomorphism lattice ec.endomorphism Endomorphism of an elliptic curve ag.endomorphism_ring Endomorphism ring ec.endomorphism_ring Endomorphism ring of an elliptic curve ec.q.endomorphism_ring Endomorphism ring of an elliptic curve cmf.eta_quotient Eta quotient ring.euclidean_domain Euclidean domain g2c.bad_lfactors Euler factors at bad primes specialfunction.gamma Euler gamma function lfunction.riemann.euler_product Euler product for the Riemann zeta function lfunction.euler_product Euler product of an L-function lfunction.history.euler Euler's work on the zeta function st_group.probabilities Event probabilities gl2.exceptional Exceptional subgroup doc.view_knowl Expand a knowl group.exponent Exponent of a group dq.ec.extent Extent of data for elliptic curves over $\Q$ dq.curve.highergenus.aut.extent Extent of data for families of higher genus curves with automorphisms dq.hecke_algebras.extent Extent of data for Hecke Algebras dq.lattice.extent Extent of data for Integral Lattices dq.rep_galois_modl.extent Extent of data for mod ℓ Galois representations dq.modlmf.extent Extent of data for mod ℓ Modular Forms dq.charmodl.extent Extent of data for mod ℓ Characters dq.ecnf.extent Extent of elliptic curve data over number fields gg.extent Extent of Galois group data nf.extent Extent of global number field data lf.extent Extent of local number field data F group.order_factorization Factorization of the group order group.representation.faithful Faithful representation ec.q.faltings_height Faltings height of an elliptic curve ec.q.faltings_ratio Faltings ratio group.families Families of abstract groups mf.siegel.family Family of Siegel groups ring.field Field gg.field_automorphisms Field automorphisms associated with a Galois group ring.field_of_fractions Field of fractions of an integral domain character.dirichlet.value_field Field of values of a Dirichlet character field.about Fields belyi.search_input Find a Belyi map mf.bianchi.search_input Find a Bianchi modular form character.dirichlet.search_input Find a Dirichlet character curve.highergenus.aut.search_input Find a family of higher genus curves gg.search_input Find a Galois group g2c.search_input Find a genus 2 curve hgm.search_input Find a hypergeometric motive cmf.search_input Find a newform or newspace by label nf.search_input Find a number field lf.search_input Find a p-adic field st_group.search_input Find a Sato-Tate group group.find_input Find an abstract group artin.search_input Find an Artin representation ec.q.search_input Find an elliptic curve ec.search_input Find an elliptic curve lattice.search_input Find an integral lattice av.fq.search_input Find an isogeny class of abelian varieties lfunction.search_input Find an L-function by label mf.hilbert.search_input Find Hilbert modular form by label st_group.first_a2_moment First $a_2$ moment group.fitting_subgroup Fitting subgroup of a group character.dirichlet.field_cut_out Fixed field of a Dirichlet character cmf.fouriercoefficients Fourier coefficients of a modular form mf.maass.picard.fourierexpansion Fourier expansion of a Maass form on PSL(2,Z[i]) mf.maass.mwf.fourierexpansion Fourier expansion of Maass form st_group.fourth_trace_moment Fourth trace moment ring.fractional_ideal Fractional ideal group.frattini_subgroup Frattini subgroup of a group ec.q.frey Frey curve cmf.fricke Fricke involution av.fq.frobenius_angles Frobenius angles nf.frobenius_cycle_types Frobenius cycle types artin.frobenius_schur_indicator Frobenius-Schur indicator of a representation group.fuchsian Fuchsian group curve.highergenus.aut.full Full group of automorphisms av.fq.class_number_1 Function fields of class number 1 lfunction.functional_equation Functional equation of an L-function group.fuchsian.fundamental_domain Fundamental domain group.sl2z.fundamental_domain Fundamental domain of $\SL(2,\Z)$ nf.fundamental_units Fundamental units of a number field dq.av.fq.further_extent Further extent of the data on isogeny classes of abelian varieties over finite fields G g2c.g2_invariants G2-invariants of a genus 2 curve nf.galois_closure Galois closure of an extension cmf.galois_conjugate Galois conjugate newforms gg.galois_group Galois group nf.galois_group Galois group nf.galois_search Galois group nf.galois_group.data Galois group information gg.label Galois group label nf.galois_group.name Galois group of a number field -- naming convention av.fq.galois_group Galois group of an isogeny class of abelian varieties lf.galois_invariants Galois invariants of a local field lf.galois_mean_slope Galois mean slope nf.galois_group.gmodule Galois module information nf.galois_group.gmodule_v4_type_i_i Galois module structure of Type I (i) for units in biquadratic number fields nf.galois_group.gmodule_v4_type_i_ii Galois module structure of Type I (ii) for units in biquadratic number fields nf.galois_group.gmodule_v4_type_i_iii Galois module structure of Type I (iii) for units in biquadratic number fields nf.galois_group.gmodule_v4_type_ii_i Galois module structure of Type II (i) for units in biquadratic number fields nf.galois_group.gmodule_v4_type_ii_ii Galois module structure of Type II (ii) for units in biquadratic number fields nf.galois_group.gmodule_v4_type_iii Galois module structure of Type III for units in biquadratic number fields nf.galois_group.gmodule_v4_type_iv Galois module structure of Type IV for units in biquadratic number fields mf.siegel.galois_orbit Galois orbit cmf.hecke_galois_orbit Galois orbit label character.dirichlet.galois_orbit Galois orbit of a Dirichlet character cmf.galois_orbit Galois orbit of a newform artin.galois_orbit Galois orbit of an Artin representation cmf.galois_representation Galois representation ec.galois_rep Galois representations attached to an elliptic curve lf.galois_splitting_model Galois splitting model of a local field lfunction.gamma_factor Gamma factors group.gassmann_equivalence Gassmann equivalence character.dirichlet.gauss_sum Gauss sum group.generalized_quaternion Generalized quaternion lfunction.rh Generalized Riemann hypothesis curve.highergenus.aut.generatingvector Generating Vector nf.field_generator Generator of a number field nf.generator Generator of a number field character.dirichlet.group.generators Generators of a Dirichlet character group group.generators Generators of a group st_group.generators Generators of a Sato-Tate group group.permutation_gens Generators of permutation group lmfdb.object_information Generic dynamic knowl g2c.g2curve Genus 2 curve belyi.genus Genus of a Belyi map modcurve.genus Genus of a modular curve ag.curve.genus Genus of a smooth curve gl2.genus Genus of a subgroup of $\GL_2$ group.sl2z.genus_subgroup Genus of a subgroup of the modular group lattice.genus Genus of an integral lattice modcurve.genus_minus_rank Genus-rank difference g2c.geom_aut_grp Geometric automorphism group ag.geom_component Geometric component modcurve.components Geometric components of a modular curve ag.geom_endomorphism_algebra Geometric endomorphism algebra g2c.geom_end_alg Geometric endomorphism algebra ag.geom_endomorphism_ring Geometric endomorphism ring ec.geom_endomorphism_ring Geometric endomorphism ring ag.gonality_geom Geometric gonality g2c.geometric_invariants Geometric invariants g2c.geom_iso_class Geometric isomorphism class of a curve ag.geom_simple Geometrically simple av.geometrically_simple Geometrically simple av.geometrically_squarefree Geometrically squarefree abelian variety belyi.geometry_type Geometry type of a Belyi map gl2.subgroup_data GL2 subgroup data ec.global_minimal_model Global minimal model ec.period Global period of an elliptic curve ag.gonality Gonality of a curve ag.good_reduction Good and bad reduction ec.good_ordinary_reduction Good ordinary reduction ec.good_reduction Good reduction ec.good_supersingular_reduction Good supersingular reduction lattice.gram Gram matrix mf.siegel.lift.gritsenko Gritsenko lift mf.sl2.subgroup.gamma0n Group $\Gamma_0(N)$ subgroup of $\textrm{SL}(2,\mathbb{Z})$ mf.sl2.subgroup.gamma1n Group $\Gamma_1(N)$ subgroup of $\textrm{SL}(2,\mathbb{Z})$ gg.group_action_invariants Group action Invariants curve.highergenus.aut.groupalgebradecomp Group algebra decomposition of Jacobian varieties group.homomorphism Group homomorphism gg.invariants Group invariants group.isomorphism Group isomorphism lattice.group_order Group order of an integral lattice group.about Groups mf.growth_condition Growth condition for a modular form gsp4.subgroup_data GSp4 subgroup data doc.knowl.annotation_guidelines Guidelines for writing annotations H group.haar_measure Haar measure of a topological group mf.half_integral_weight Half-integral weight modular forms group.subgroup.hall Hall subgroup g2c.has_square_sha Has square Ш combin.hasse_diagram Hasse diagram g2c.hasse_weil_conjecture Hasse-Weil conjecture lfunction.history.hasse_weil Hasse-Weil L-function hecke_algebra.definition Hecke algebra mf.bianchi.hecke_algebra Hecke algebra character.hecke Hecke character cmf.heckecharpolys Hecke characteristic polynomial mf.gl2.history.hecke Hecke Characters mf.hilbert.eigenvalue Hecke eigenvalues cmf.hecke_kernels Hecke kernel lfunction.history.hecke_characters Hecke L-functions cmf.hecke_operator Hecke operator lfunction.history.hecke_operators Hecke operators, history cmf.hecke_orbit Hecke orbit mf.hilbert.hecke_orbit Hecke orbit for Hilbert modular form lattice.hermite_number Hermite number hgm.field.label HGM Labels crystals.highest_weight Highest weight crystals mf.hilbert Hilbert modular form nf.ideal.label.hmf Hilbert modular form convention for the label of a number field ideal lattice.history History of Lattices st_group.hodge_circle Hodge circle mot.hodgevector Hodge vector of a hypergeometric motive cmf.eisenstein Holomorphic Eisenstein series of level 1 av.fq.honda_tate Honda-Tate theorem rcs.cite.av.fq How to cite abelian varieties over finite fields rcs.cite.groups.abstract How to cite abstract groups rcs.cite.artin How to cite Artin representations rcs.cite.belyi How to cite Belyi maps rcs.cite.mf.bianchi How to cite Bianchi modular forms rcs.cite.cmf How to cite classical modular forms rcs.cite.curve.highergenus.aut How to cite curves of higher genus with automorphism group data rcs.cite.character.dirichlet How to cite Dirichlet characters rcs.cite.ec.q How to cite elliptic curves over $\mathbb{Q}$ rcs.cite.gg How to cite Galois groups rcs.cite.g2c How to cite genus 2 curves rcs.cite.mf.hilbert How to cite Hilbert modular forms rcs.cite.hgm How to cite hypergeometric motive data rcs.cite.lfunction How to cite L-functions rcs.cite.maass How to cite Maass forms rcs.cite.nf How to cite number fields rcs.cite.lf How to cite p-adic fields rcs.cite.st_group How to cite Sato-Tate groups content.how_to_cite How to cite the LMFDB dq.zeros.zeta.howcomputed How zeros of the Riemann zeta function were computed curve.highergenus.hurwitz_bound Hurwitz bound mf.bianchi.hyperbolic3space Hyperbolic 3-space group.hyperelementary Hyperelementary group ag.hyperelliptic_curve Hyperelliptic curve hgm.familes Hypergeometric family over $\Q$ hgm.motive Hypergeometric motive I nf.ideal_class_group Ideal class group of a number field nf.ideal_labels Ideal labels ring.ideal Ideal of a ring doc.knowl.identifier Identifier of a knowl st_group.identity_component Identity component of a Sato-Tate group alg.binary_operation.identity Identity for a binary operation g2c.igusa_invariants Igusa invariants of a genus 2 curve g2c.igusa_clebsch_invariants Igusa-Clebsch invariants of a genus 2 curve ec.galois_rep_elladic_image Image of $\ell$-adic Galois representation g2c.galois_rep_image Image of mod p Galois Representation of Jacobian of genus 2 curve ec.galois_rep_modell_image Image of mod-$\ell$ Galois representation artin.gg_quotient Image of the representation doc.knowl.implicit Implicit creation of knowls hgm.imprimitivity_index Imprimitivity index of a hypergeometric motive cmf.include_all_spaces Include all spaces gg.int_modules Indecomposable integral representation modcurve.index Index of a modular curve nf.zk_index Index of a number field group.subgroup.index Index of a subgroup gl2.index Index of an open subgroup nf.index Index of an order st_group.index Index of the identity component intro.description Index page, first sentences lf.indices_of_inseparability Indices of inseparability character.dirichlet.induce Induced Dirichlet character lf.inertia_group Inertia group lf.inertia_group_search Inertia group searching nf.inessential_prime Inessential prime mf.gl2.history.infinite Infinite products lfunction.infinity_factor Infinity factor of an L-function users.jhs.note Information about Joe Silverman av.fq.initial_coefficients Initial coefficients of the L-polynomial group.inner_automorphism Inner automorphism of a group cmf.inner_twist Inner twist cmf.inner_twist_count Inner twist count cmf.inner_twist_group Inner twist group cmf.inner_twist_multiplicity Inner twist multiplicity cmf.inner_twist_proved Inner twist proved gg.arithmetically_equiv_input Input for arithmetically equivalent fields nf.integral_basis Integral basis of a number field ring.integral_domain Integral domain ring.integral Integral element of a ring nf.integral Integral elements lattice.definition Integral Lattice ec.integral_model Integral model ec.q.integral_points Integral points ring.integrally_closed Integrally closed group.properties_interdependencies Interdependencies of group properties group.subgroup_properties_interdependencies Interdependencies of subgroup properties lf.intermediate_fields Intermediate fields nf.intermediate_fields Intermediate fields intro Introduction intro.features Introduction/Features g2c.invariants Invariants of a genus 2 curve modcurve.invariants Invariants of a modular curve lfunction.invariants Invariants of an L-function alg.binary_operation.inverse Inverse for a binary operation ring.irreducible Irreducible element repn.irrep.gl Irreducible representations of $\GL(g,\C)$ ag.irreducible Irreducible variety nf.is_galois Is a Galois extension group.isoclinism Isoclinism ec.isogeny Isogeny between elliptic curves av.fq.abvar.data Isogeny class data ec.isogeny_class_degree Isogeny class degree av.isogeny_class Isogeny class of an abelian variety g2c.isogeny_class Isogeny class of an abelian variety ec.isogeny_class Isogeny class of an elliptic curve modcurve.decomposition Isogeny decomposition of the Jacobian of a modular curve ec.isogenygraph.legend Isogeny graph legend ec.isogeny_graph Isogeny graph of an isogeny class of elliptic curves ec.isogeny_matrix Isogeny matrix of an isogeny class of elliptic curves av.isogeny Isogeny of abelian varieties nf.isogeny_primes Isogeny primes of elliptic curves lattice.isometry Isometries between integral lattices ec.weierstrass_isomorphism Isomorphism between Weierstrass models av.fq.isogeny_class_size Isomorphism classes within an isogeny class ec.isomorphism Isomorphism of elliptic curves ec.iwasawa_invariants Iwasawa invariants of elliptic curves J ec.j_invariant_denominator_valuation j-invariant denominator valuation ec.q.j_invariant j-invariant of a rational elliptic curve ec.j_invariant j-invariant of an elliptic curve mf.jacobi Jacobi forms character.dirichlet.jacobi_sum Jacobi sum character.dirichlet.jacobi_symbol Jacobi symbol ag.jacobian Jacobian of a curve av.fq.jacobian Jacobian of a curve g2c.jacobian Jacobian of a genus 2 curve K lattice.kissing Kissing number character.dirichlet.kloosterman_sum Kloosterman sum doc.knowl Knowl doc.knowl.guidelines Knowl Edit Guidelines test.knowlparam Knowl with parameters g2c.known_rational_points Known rational points ec.kodaira_symbol Kodaira symbol ec.q.kodaira_symbol Kodaira symbol mf.siegel.koecher.principle Koecher principle character.dirichlet.kronecker_symbol Kronecker symbol L lfunction L-function cmf.lfunction L-function of a classical modular form g2c.lfunction L-function of an abelian variety over $\mathbb{Q}$ lfunction.spectral_parameters L-function spectral parameters lfunction.history.maass L-functions of Maass forms av.fq.l-polynomial L-polynomial of an abelian variety lf.field.label Label for a $p$-adic field artin.label Label for an Artin representation ec.q.lmfdb_label Label for an elliptic curve over $\mathbb Q$ cmf.label Label of a classical modular form group.label_conjugacy_class Label of a conjugacy class character.dirichlet.galois_orbit_label Label of a Galois orbit of a Dirichlet character g2c.label Label of a genus 2 curve over $\Q$ group.label Label of a group modcurve.label Label of a modular curve mf.siegel.label Label of a Siegel modular form mf.hilbert.label Label of an Hilbert Modular Form group.label_complex_group_char Label of irreducible complex characters of groups lfunction.label Label of L-function group.label_rational_group_char Label of rational character of a group dq.curve.highergenus.aut.label Labeling convention for families of higher genus curves character.dirichlet.conrey.orbit_label Labels for Galois orbits of Dirichlet Characters av.fq.lmfdb_label Labels for isogeny classes of abelian varieties over finite fields group.small_group_label Labels for the small group database in GAP and Magma mf.bianchi.labels Labels of Bianchi modular forms ec.curve_label Labels of elliptic curves over number fields ec.lambda_invariant lambda-invariant of an elliptic curve lfunction.history.langlands_program Langlands program mf.maass.mwf.laplacian Laplace-Beltrami operator (hyperbolic) mf.maass.picard.laplacian Laplacian in 3d hyperbolic upper half-space doc.macros LaTeX macros in the LMFDB lfunction.leading_coeff Leading coefficient lfunction.degree2holo.key Legend for GL(2) holomorphic graph of dots character.dirichlet.legendre_symbol Legendre symbol mf.bianchi.level Level of a Bianchi modular form mf.hilbert.level_norm Level of a Hilbert modular form modcurve.level Level of a modular curve cmf.level Level of a modular form sh.level Level of a Shimura curve mf.siegel.level Level of a Siegel modular form lattice.level Level of an integral lattice gl2.level Level of an open subgroup modcurve.level_structure Level structure of a modular curve hgm.levelt_matrices Levelt matrices of a hypergeometric family mf.siegel.lift Lift Siegel modular form group.linear Linear and Projective Linear Groups crystals.littelmann_path Littelmann path model for crystals doc.lmfdb.improve LMFDB backend lattice.label LMFDB label of an integral lattice doc.news.in_the_news LMFDB Press lf.algebra.data Local algebra nf.local_algebra Local algebra ec.local_data Local data of an elliptic curve lf.local_field Local field lf.invariants Local field invariants g2c.local_invariants Local invariants g2c.good_lfactors Local L-factors at good primes ec.local_minimal_discriminant Local minimal discriminant of an elliptic curve ec.local_minimal_model Local minimal model ec.local_root_number Local root number g2c.locally_solvable Locally solvable group.lower_central_series Lower central series of a group lucant LuCaNT Conference M mf.maass.mwf.coefficients Maass form coefficients mf.maass.mwf Maass form on GL2(R) with trivial character mf.maass.mwf.plot Maass form plot mf.maass.mwf.weight0 Maass forms of weight 0 mf.maass Maass forms on GL(n) mf.maass.picard Maass forms on PSL(2,Z[i]) mf.siegel.lift.maass Maass lift from Jacobi form to Siegel modular form content.edit_board Management board ec.q.manin_constant Manin constant for elliptic curves over $\Q$ ec.maximal_elladic_galois_rep Maximal $\ell$-adic Galois representation ec.maximal_ladic_galois_rep Maximal $\ell$-adic Galois representation ec.maximal_galois_rep Maximal Galois representation ring.maximal_ideal Maximal ideal lf.maximal_ideal Maximal ideal of a local field group.maximal_quotient Maximal quotient of a group group.maximal_subgroup Maximal subgroup of a group st_group.subgroups Maximal subgroups group.metabelian Metabelian group group.metacyclic Metacyclic group character.dirichlet.minimal Minimal Dirichlet character ec.minimal_discriminant Minimal discriminant g2c.minimal_equation Minimal equation of a hyperelliptic curve ag.minimal_field Minimal field of definition cmf.minimal Minimal modular form group.minimal_normal Minimal normal subgroup nf.minimal_polynomial Minimal polynomial st_group.supgroups Minimal supergroups lattice.minimal_vector Minimal vector length ec.q.minimal_weierstrass_equation Minimal Weierstrass equation over $\mathbb Q$ cmf.minus_space Minus space mf.siegel.lift.miyawaki Miyawaki lift mf.siegel.lift.miyawaki1 Miyawaki lift of type I, Ikeda-Miyawaki lift mf.siegel.lift.miyawaki2 Miyawaki lift of type II modcurve.modular_cover Modular cover modcurve Modular curve modcurve.xn Modular curve $X(N)$ ag.modcurve.x0 Modular curve $X_0(N)$ ag.modcurve.x1 Modular curve $X_1(N)$ ag.modcurve.xnsp Modular curve $X_{\text{ns}}^{+}(N)$ ec.q.modular_degree Modular degree of an elliptic curve over $\mathbb Q$ ec.q.modular_form Modular form associated to an elliptic curve/$\mathbb Q$ mf.about Modular Forms ec.q.modular_parametrization Modular parametrization of an elliptic curve over $\mathbb Q$ character.dirichlet.modulus Modulus of a Dirichlet character st_group.moment_matrix Moment matrix st_group.moment_simplex Moment simplex st_group.moments Moment statistics curve.highergenus.aut.monodromy Monodromy Group belyi.group Monodromy group of a Belyi map hgm.monodromy Monodromy group of a hypergeometric family nf.monogenic Monogenic field group.monomial Monomial group nf.monomial_order Monomial order g2c.mw_generator Mordell-Weil generator ec.mw_generators Mordell-Weil generators ec.mordell_weil_group Mordell-Weil group ag.mordell_weil Mordell-Weil group of an abelian variety g2c.mordell_weil_rank Mordell-Weil rank ec.mordell_weil_theorem Mordell-Weil theorem motives.about Motives lfunction.motivic_weight Motivic weight ec.mu_invariant mu-invariant of an elliptic curve af.multiplicative Multiplicative arithmetic function nf.multiplicative_gal_module Multiplicative Galois Module Structure ec.multiplicative_reduction Multiplicative reduction mf.maass.mwf.dimension Multiplicity of an eigenvalue mf.multipliersystem Multiplier system group.mobius_function Möbius function of subgroup lattice N lattice.name Name of a lattice st_group.name Name of a Sato-Tate group doc.knowl.naming_conventions Naming conventions for knowls doc.knowl.naming_suggestions Naming suggestions nf.narrow_class_group Narrow class group nf.narrow_class_number Narrow class number mf.gl2.history.new New modular forms from given ones cmf.newform Newform modcurve.newform_level Newform level of a modular form cmf.newform_subspace Newform subspace mf.siegel.newform_subspace Newform subspace lf.newton_polygon Newton polygon of a polynomial group.nilpotent Nilpotent group cmf.nk2 Nk^2 ring.noetherian Noetherian ring ag.singular_point Non-singular point (definition) gl2.nonsplit_cartan Non-split Cartan subgroup ec.nonsplit_multiplicative_reduction Non-split multiplicative reduction cmf.nontrivial_twist Nontrivial inner twist ec.discriminant_norm Norm of discriminant group.subgroup.normal_closure Normal closure of a subgroup group.normal_series Normal series of a group group.subgroup.normal Normal subgroup of a group lfunction.normalization Normalization of an L-function nf.defining_polynomial.normalization Normalization of defining polynomials for number fields lattice.normalized_minimal_vector Normalized minimal vector of a lattice gl2.normalizer_cartan Normalizer of a Cartan subgroup gl2.normalizer_nonsplit_cartan Normalizer of a non-split Cartan subgroup gl2.normalizer_split_cartan Normalizer of a split Cartan subgroup group.subgroup.normalizer Normalizer of a subgroup artin.number_field Number field associated to an Artin representation nf.invariants Number field invariants nf.label Number field label nf.nickname Number field nicknames nf.field.missing Number field not in the database av.fq.number_field Number field of an isogeny class of abelian varieties character.dirichlet.related_fields Number fields related to a Dirichlet character belyi.num_orbits Number of Galois orbits av.hyperelliptic_count Number of hyperelliptic Jacobians av.jacobian_count Number of Jacobians g2c.num_rat_pts Number of rational points O ec.obstruction_class Obstruction class of an elliptic curve cmf.oldspace Old subspace of modular forms cmf.decomposition.old.gamma1 Oldspace decomposition for Gamma1(N) cmf.decomposition.old.gamma0chi Oldspace decomposition for newform space gl2.open Open subgroup ec.q.optimal Optimal elliptic curve over $\mathbb Q$ character.dirichlet.galois_orbit_index Orbit index of a Dirichlet character belyi.orbit_size Orbit size nf.order Order character.dirichlet.order Order of a Dirichlet character character.dirichlet.conrey.order Order of a Dirichlet character from its label character.dirichlet.group.order Order of a Dirichlet character group group.order Order of a group group.element_order Order of a group element g2c.mw_generator_order Order of a Mordell-Weil generator group.order_stats Order statistics of a group av.fq.ordinary Ordinary modcurve.other_labels Other labels for modular curves group.outer_aut Outer automorphism group group.over_subgroup Over-subgroup P hgm.defining_parameter_ppart p-part of defining parameters of a hypergeometric motive av.fq.p_rank p-rank g2c.paramodular_conjecture Paramodular conjecture group.paramodular Paramodular group mf.siegel.newform.paramodular Paramodular newform character.dirichlet.parity Parity of a Dirichlet character character.dirichlet.conrey.parity Parity of a Dirichlet character from its Conrey label gg.parity Parity of a Galois group artin.parity Parity of a representation belyi.passport Passport belyi.pass_size Passport size of a Belyi map group.perfect Perfect groups ec.q.period_lattice Period lattice of an elliptic curve group.permutation_degree Permutation degree group.permutation_representation Permutation representation cmf.petersson_scalar_product Petersson scalar product nf.place Place of a number field ag.plane_model Plane model cmf.plus_space Plus space mf.gl2.history.poincare Poincaré series specialfunction.poincare Poincaré series ag.fq.point_counts Point counts of a variety av.fq.curve_point_counts Point counts of the curve av.polarization Polarization lattice.postive_definite Positive definite lattice ag.potential_good_reduction Potential good reduction ec.potential_good_reduction Potential good reduction av.potential_toric_rank Potential toric rank dq.cmf.cc_prec Precision of classical modular form eigenvalues group.presentation Presentation of a finite group doc.lmfdb.contextualize Presentation of the data in the LMFDB group.primary_decomposition Primary decomposition of abelian group ring.prime_ideal Prime ideal nf.prime Prime of a number field ag.bad_prime Prime of bad reduction ag.good_prime Prime of good reduction hgm.defining_parameter_primetoppart prime to $p$-part of defining parameters of a hypergeometric motive g2c.good_reduction Primes of good reduction ag.primitive Primitive algebraic variety belyi.primitive Primitive Belyi map character.dirichlet.primitive Primitive Dirichlet character nf.primitive_element Primitive element lattice.primitive Primitive integral lattice lfunction.primitive Primitive L-function gg.primitive Primitivity of a Galois group belyi.primitivization Primitivization of a Belyi map character.dirichlet.principal Principal Dirichlet character ring.principal_fractional_ideal Principal fractional ideal ag.torsor Principal homogeneous space ring.principal_ideal_domain Principal ideal domain av.princ_polarizable Principally polarizable gl2.profinite Profinite group artin.projective_field Projective Artin field cmf.projective_field Projective field cmf.projective_image Projective image group.subgroup.projective_image Projective image of a subgroup artin.projective_image Projective image of an Artin representation artin.projective_image_type Projective image type ag.projective_space Projective space artin.projective_stem_field Projective stem field of an Artin representation group.proper_subgroup Proper subgroup stats.proportions Proportions for statistics Q mf.hilbert.q_expansion q-expansion of a Hilbert modular form cmf.q-expansion q-expansion of a modular form mf.quasi_modular Quasi-modular form group.quasisimple Quasisimple group ag.quotient_curve Quotient curve curve.highergenus.aut.quotientgenus Quotient genus group.quotient Quotient group group.quotient_isolabel Quotient label R group.radical Radical of a group lfunction.history.ramanujan_tau Ramanujan tau L-function lf.ramification_index Ramification index of a prime in a local field lf.ramification_polygon Ramification polygon belyi.ramification_type Ramification type nf.ramified_primes Ramified (rational) prime of a number field artin.ramified_primes Ramified prime of an Artin representation group.rank Rank nf.rank Rank of a number field lfunction.history.rankin_selberg Ranking-Selberg convolution, history group.representation.rational_character Rational character group.rational_character_table Rational character table of a group group.rational_group Rational group lfunction.rational Rational L-function g2c.all_rational_points Rational points modcurve.rational_points Rational points on a modular curve st_group.rational Rational Sato-Tate group st_group.real_dimension Real dimension of a Sato-Tate group character.dirichlet.real Real Dirichlet character nf.real_embedding Real embedding ag.real_endomorphism_algebra Real endomorphism algebra ag.real_geom_endomorphism_algebra Real geometric endomorphism algebra ag.real_multiplication Real multiplication cmf.rm_form Real multiplication ec.q.real_period Real period g2c.real_period Real period ec.reduction Reduction of an elliptic curve ec.reduction_type Reduction type ec.q.reduction_type Reduction type of an elliptic curve over $\mathbb Q$ curve.highergenus.aut.refinedpassport Refined passport g2c.regulator Regulator of a genus 2 curve nf.regulator Regulator of a number field ag.regulator Regulator of an abelian variety ec.regulator Regulator of an elliptic curve ec.q.regulator Regulator of elliptic curve doc.related_to Related objects cmf.relative_dimension Relative dimension rcs.release.1.3 Release notes for version 1.3 rcs.rigor.groups.abstract Reliability of abstract groups data rcs.rigor.artin Reliability of Artin representation data rcs.rigor.belyi Reliability of Belyi map data dq.mf.bianchi.reliability Reliability of Bianchi Modular Form data rcs.rigor.mf.bianchi Reliability of Bianchi modular form data rcs.rigor.lfunction.curve Reliability of Curve L-function Data rcs.rigor.character.dirichlet Reliability of Dirichlet Character Data rcs.rigor.lfunction.dirichlet Reliability of Dirichlet L-function Data rcs.rigor.lfunction.lcalc Reliability of Dynamically Computed L-function Data rcs.rigor.ec Reliability of elliptic curve data rcs.rigor.ec.q Reliability of elliptic curve data over $\Q$ dq.ecnf.reliability Reliability of elliptic curve data over number fields rcs.rigor.lfunction.ec Reliability of Elliptic Curve L-function Data rcs.rigor.g2c Reliability of Genus 2 Curve Data rcs.rigor.nf Reliability of global number field data rcs.rigor.mf.hilbert Reliability of Hilbert modular form data rcs.rigor.hgm Reliability of hypergeometric motive data rcs.rigor.lfunction Reliability of L-function data rcs.rigor.lattice Reliability of lattice data rcs.rigor.lf Reliability of local field data rcs.rigor.lfunction.maass Reliability of Maass form L-function data rcs.rigor.maass Reliability of Maass forms data rcs.rigor.modcurve Reliability of modular curve data rcs.rigor.lfunction.modular Reliability of Modular L-function Data rcs.rigor.zeros.zeta Reliability of Riemann Zeta Zeros Data rcs.rigor.st_group Reliability of Sato-Tate Group Data rcs.rigor.cmf Reliability of the Classical Modular Form Data dq.ec.reliability Reliability of the elliptic curve data dq.curve.highergenus.aut.reliability Reliability of the higher genus curves with automorphisms data rcs.rigor.curve.highergenus.aut Reliability of the higher genus curves with automorphisms data dq.mf.hilbert.reliability Reliability of the Hilbert modular form data rcs.rigor.gg Reliability of transitive group data rcs.rigor.av.fq Reliability of Weil polynomial data mf.gl2.history.remainder Remainder terms doc.knowl.rename Renaming a knowl repn.about Representation lf.residue_field Residue field lf.residue_field_degree Residue field degree of a local field gg.resolvents Resolvents of a Galois group ag.riemann_surface Riemann surface lfunction.riemann Riemann zeta-function lfunction.history.riemanns_memoir Riemann's memoir lf.ring_of_integers Ring of integers of a local field nf.ring_of_integers Ring of integers of a number field character.dirichlet.roadmap Roadmap for Dirichlet characters ec.roadmap Roadmap for elliptic curves content.roadmap Roadmap for future areas of LMFBD mf.hilbert.roadmap Roadmap for Hilbert modular forms lfunction.roadmap Roadmap for L-functions lf.roadmap Roadmap for local number fields mf.maass.roadmap Roadmap for Maass forms mf.siegel.roadmap Roadmap for Siegel modular forms cmf.root Root lfunction.root_analytic_conductor Root analytic conductor lfunction.root_angle Root angle of L-function lfunction.root_angle_input Root angle of L-function nf.root_discriminant Root discriminant of a number field lattice.root_lattice Root lattice lf.root_number Root number of a local field artin.root_number Root number of an Artin representation ec.rouse_label Rouse label S sage.test Sage cell knowl mf.siegel.lift.saito_kurokawa Saito-Kurokawa lift cmf.satake_parameters Satake parameters st_group.definition Sato-Tate group ag.sato_tate_group Sato-Tate group (of a motive) st_group.data Sato-Tate group data g2c.st_group_identity_component Sato-Tate group identity component st_group.invariants Sato-Tate group invariants st_group.label Sato-Tate group labels g2c.st_group Sato-Tate group of a genus 2 curve over Q cmf.sato_tate Sato-Tate group of a modular form st_group.subsupgroups Sato-Tate subgroups and supergroups group.schur_multiplier Schur multiplier intro.search Search nf.padic_completion.search Searching $p$-adic completions of a number field ec.galois_image_search Searching on Galois images st_group.second_trace_moment Second trace moment lfunction.selberg_class.axioms Selberg class axioms lfunction.selberg_class Selberg class of L-functions lfunction.selbergdata Selberg data for the functional equation of an L-function doc.select_search_columns Select search columns cmf.selfdual Self dual modular form cmf.self_twist_field Self twist field mf.ellitpic.self_twist Self twist of a modular form lfunction.self-dual Self-dual L-function cmf.self_twist Self-twist ag.selmer_group Selmer group of an abelian variety ec.semi_global_minimal_model Semi-global minimal model av.semiabelian_variety Semiabelian variety group.semidirect_product Semidirect product of groups ec.q.semistable Semistable elliptic curve ec.semistable Semistable elliptic curve nf.separable_algebra Separable algebra nf.separable Separable extension nf.serre_odlyzko_bound Serre Odlyzko bound cmf.shimura_correspondence Shimura correspondence mf.half_integral_weight.shimura_decomposition Shimura decomposition ag.shimura_variety Shimura variety nf.sibling Sibling fields and algebras gg.other_representations Siblings for a Galois group mf.siegel.phi Siegel $\Phi$ operator mf.siegel.cusp_form Siegel cusp form mf.siegel.cusp_form_degree2 Siegel cusp form of degree 2 mf.siegel.eisenstein_series Siegel Eisenstein series mf.siegel.klingen_eisenstein_series Siegel Klingen-Eisenstein series mf.siegel Siegel modular form mf.siegel.hecke_operator Siegel modular form Hecke operators mf.siegel.character Siegel modular form with character lfunction.history.siegel Siegel modular forms, history mf.siegel.newform Siegel newform mf.siegel.theta_series Siegel theta series and theta series with pluriharmonics mf.siegel.halfspace Siegel upper half-space mf.bianchi.sign Sign of a Bianchi newform lfunction.root_number Sign of an L-function lfunction.sign Sign of the functional equation curve.highergenus.aut.signature Signature of a group action on a curve nf.signature Signature of a number field lfunction.signature Signature of an L-function ag.simple Simple av.simple Simple abelian variety gg.simple_name Simple group name group.simple Simple groups modcurve.simple Simple modular curve ec.simple_equation Simplified equation g2c.simple_equation Simplified equation lf.slope_content Slope content of a $p$-adic field lf.newton_slopes Slopes of the Newton polygon of a polynomial ag.curve.smooth Smoothness of an algebraic curve group.socle Socle of a group group.solvable Solvable group cmf.sort_order Sort order group.sort_order Sort order for groups rcs.source.groups.abstract Source of abstract group data rcs.groups.abstract.source Source of abstract group data rcs.source.artin Source of Artin representation data dq.belyi.source Source of Belyi maps data rcs.source.belyi Source of Belyi maps data dq.mf.bianchi.source Source of Bianchi modular form data rcs.source.mf.bianchi Source of Bianchi modular form data rcs.source.cmf Source of Classical Modular Forms Data rcs.source.character.dirichlet Source of Dirichlet character data rcs.source.lfunction.dirichlet Source of Dirichlet L-function Data rcs.source.lfunction.lcalc Source of Dynamically Computed L-function Data rcs.source.ec Source of elliptic curve data dq.ec.source Source of elliptic curve data over $\Q$ rcs.source.ec.q Source of elliptic curve data over $\Q$ dq.ecnf.source Source of elliptic curve data over number fields rcs.source.lfunction.ec Source of Elliptic Curve L-function Data rcs.source.g2c Source of Genus 2 Curve Data rcs.source.nf Source of global number field data rcs.source.curve.highergenus.aut Source of higher genus curves with automorphisms data dq.mf.hilbert.source Source of Hilbert modular form data rcs.source.mf.hilbert Source of Hilbert modular form data dq.hgm.source Source of hypergeometric data rcs.source.hgm Source of hypergeometric data rcs.source.lfunction Source of L-function data rcs.source.lfunction.curve Source of L-function Data dq.lattice.source Source of lattice data rcs.source.lattice Source of lattice data rcs.source.lfunction.maass Source of Maass form L-function Data rcs.source.maass Source of Maass forms data rcs.source.modcurve Source of modular curve data rcs.source.lfunction.modular Source of Modular L-function Data rcs.source.lf Source of p-adic field data rcs.source.zeros.zeta Source of Riemann Zeta Zeros Data dq.st.source Source of Sato-Tate group data rcs.source.st_group Source of Sato-Tate Group Data dq.curve.highergenus.aut.source Source of the higher genus curves with automorphisms data rcs.source.gg Source of transitive group data cmf.newspace Space of newforms cmf.newspaces Space of newforms mf.siegel.newspace Space of paramodular newforms mf.bianchi.spaces Spaces of Bianchi modular forms cmf.space Spaces of modular forms ec.q.special_value Special value of an elliptic curve L-function lfunction.spectral_label Spectral label mf.maass.picard.eigenvalue Spectral parameter of Maass forms on PSL(2,Z[i]) group.nonsplit_product Split and non-split extension of groups gl2.split_cartan Split Cartan subgroup ec.split_multiplicative_reduction Split multiplicative reduction ag.splitting_field Splitting field av.squarefree Squarefree abelian variety curve.highergenus.aut.generators Standard generators modcurve.name Standard modular curves modcurve.standard Standard modular curves group.stem_extension Stem extension nf.stem_field Stem field for a Galois extension group.stem_group Stem group character.dirichlet.group.structure Structure of a Dirichlet group cmf.sturm_bound Sturm bound cmf.sturm_bound_gamma1 Sturm bound for $\Gamma_1(N)$ gg.subfields Subfields group.sp2gr.ppal.congruence Subgroup $\Gamma(N)$ of $\Sp(2g,\Z)$ group.sl2z.subgroup.gamman Subgroup $\Gamma(N)$ of $\textrm{SL}(2,\mathbb{Z})$ group.sp2gr.gamma0 Subgroup $\Gamma_0(N)$ of $\Sp(2g,\Z)$ group.sl2z.subgroup.gamma0n Subgroup $\Gamma_0(N)$ of $\textrm{SL}(2,\mathbb{Z})$ group.sp2gr.gamma1 Subgroup $\Gamma_1(N)$ of $\Sp(2g,\Z)$ group.sl2z.subgroup.gamma1n Subgroup $\Gamma_1(N)$ of $\textrm{SL}(2,\mathbb{Z})$ group.subgroup.diagram Subgroup diagram group.subgroup.diagram.lmfdb Subgroup diagrams in the LMFDB group.subgroup_isolabel Subgroup label group.subgroup_label Subgroup labels group.subgroup Subgroup of a group gg.subnormal_series Subnormal series mf.siegel.subspace.degree.two Subspaces of degree 2 Siegel modular forms cmf.subspaces Subspaces of modular forms av.fq.supersingular Supersingular abelian variety group.supersolvable Supersolvable group group.sylow_subgroup Sylow subgroup group.symmetric Symmetric group lfunction.symm Symmetric power of an L-function st_group.symplectic_form Symplectic form group.sp2gr Symplectic group alg.symplectic_isomorphism Symplectic isomorphism T zeros.first.contents Table of first zeros of L-function ec.tamagawa_number Tamagawa number g2c.tamagawa Tamagawa numbers ec.q.tamagawa_numbers Tamagawa numbers for an elliptic curve lf.tame_degree Tame degree of a local field extension ec.padic_tate_module Tate module of an elliptic curve lfunction.history.tates_thesis Tate's thesis ag.tate_shafarevich Tate-Shafarevich group lfunction.taxonomy Taxonomy of L-functions test.text test users.edgarcosta.note test test.suggestions Test knowl suggestions mf.maass.mwf.eigenvalue The eigenvalue of a Maass form mf.maass.mwf.level The level of a Maass form mf.maass.mwf.ncoefficients The number of coefficients mf.maass.mwf.precision The precision of an eigenvalue mf.siegel.family.kp The ring of paramodular cusp forms $S_k(K(p))$ mf.siegel.family.gamma_2 The ring of Siegel modular forms of degree 2 for $\Gamma(2)$ mf.siegel.family.gamma0_2 The ring of Siegel modular forms of degree 2 for $\Gamma_0(2)$ mf.siegel.family.gamma0_3 The ring of Siegel modular forms of degree 2 with respect to $\Gamma_0(3)$ mf.siegel.family.gamma0_3_psi_3 The ring of Siegel modular forms of degree 2 with respect to $\Gamma_0(3)$ with character $\psi_3$ mf.siegel.family.gamma0_4 The ring of Siegel modular forms of degree 2 with respect to $\Gamma_0(4)$ mf.siegel.family.gamma0_4_psi_4 The ring of Siegel modular forms of degree 2 with respect to $\Gamma_0(4)$ with character mf.siegel.family.sp4z The ring of Siegel modular forms of degree 2 with respect to the full modular group mf.siegel.family.sp6z The ring of Siegel modular forms of degree 3 with respect to the full modular group mf.siegel.family.sp8z The ring of Siegel modular forms of degree 4 with respect to the full modular group mf.maass.mwf.spectralparameter The spectral parameter of a Maass waveform mf.maass.mwf.symmetry The symmetry of a Maass form lf.unramified_totally_ramified_tower The unramified/totally ramified tower of a local field mf.maass.mwf.weight The weight of a Maass form mf.siegel.theta_constant Theta constant av.theta_divisor Theta divisor on a principally polarized abelian variety mf.half_integral_weight.theta Theta function mf.gl2.history.theta Theta functions lattice.theta Theta series for an integral lattice dq.zeros.zeta.toomany Too many zeros lf.top_slope Top slope for a local field curve.highergenus.aut.topological_equivalence Topological Equivalence group.torsion Torsion group ec.q.torsion_growth Torsion growth in number fields g2c.torsion_order Torsion order ec.torsion_order Torsion order of an elliptic curve g2c.torsion Torsion subgroup ec.torsion_subgroup Torsion subgroup of an elliptic curve ec.q.torsion_subgroup Torsion subgroup of an elliptic curve over $\mathbb Q$ nf.totally_imaginary Totally imaginary nf.totally_positive Totally positive nf.totally_real Totally real stats.totals Totals for statistics cmf.trace_bound Trace bound cmf.space_trace_form Trace form cmf.trace_form Trace form st_group.trace_moments Trace moments of a Sato-Tate group artin.trace_of_complex_conj Trace of complex conjugation st_group.trace_zero_density Trace zero density of a Sato-Tate group doc.knowl.context_free Transclusion (definition) mf.transformation_property Transformation property for modular forms gg.conway_name Transitive group name conventions belyi.permutation_triple Transitive permutation triple group.triangle_group Triangle group group.trivial_subgroup Trivial subgroup lfunction.trivial_zero Trivial zeros of an L-function nf.sextic_twin Twin sextic algebra cmf.twist Twist cmf.twist_minimal Twist minimal cmf.twist_multiplicity Twist multiplicity av.twist Twist of an abelian variety group.representation.type Type of a complex group representation group.type Type of groups U group.under_subgroup Under-subgroup lfunction.underlying_object Underlying object of an L-function character.unit_group Underlying unit group for a Dirichlet character lattice.unimodular Unimodular lattice ring.unique_factorization_domain Unique factorization domain nf.unit_group Unit group of a number field nf.torsion Unit group torsion ring.unit Unit in a ring st_group.usp Unitary symplectic group nf.unramified_prime Unramified (rational) prime of a number field lf.unramified_degree Unramified degree of a local field artin.unramified_primes Unramified prime of an Artin representation lf.unramified_subfield Unramified subfield doc.upload_data Upload data group.upper_central_series Upper central series of a group mf.upper_half_plane Upper half-plane V character.dirichlet.values Values of a Dirichlet character character.dirichlet.values_on_gens Values of a Dirichlet character on generators mf.siegel.family.sp4z_2 Vector valued Siegel modular forms of degree 2 taking values in three-dimensional space mf.siegel.vector_valued Vector-valued Siegel modular forms mf.siegel.vector_valued_degree_two Vector-valued Siegel modular forms of degree 2 lf.visible_slopes Visible slopes W ec.weierstrass_coeffs Weierstrass equation or model g2c.num_rat_wpts Weierstrass points of a hyperelliptic curve hgm.weight weight mf.siegel.weight_k_j Weight mf.slash_action Weight $k$ slash-action mf.bianchi.weight Weight of a Bianchi modular form mf.hilbert.weight_vector Weight of a Hilbert modular form st_group.weight Weight of a Sato-Tate group mf.siegel.weight Weight of a Siegel modular form cmf.weight Weight of an elliptic modular form nf.weil_height Weil height ec.q.weil_height Weil height of a point on an elliptic curve over $\Q$ nf.weil_polynomial Weil polynomial group.weyl_group Weyl group group.fuchsian.cusps.width Width of a cusp mot.width Width of a motive doc.wiki Wiki (definition) lf.wild_inertia_group Wild inertia group lf.wild_inertia_group_search Wild inertia group search lf.wild_slopes Wild slopes of a local field group.wreath_product Wreath product Z lfunction.zfunction Z-function of an L-function group.z_group Z-group ring.zero_divisor Zero divisor lfunction.zeros Zeros of an L-function ? content.news curve.highergenus.aut.sort_order mf.bianchi.fourierbessell mf.bianchi.weight2 rcs.ack.lattice sage.cell users.kedlaya.note users.lewismcombes.note users.pascal.note g2c.gl2type $\GL_2$-type ec.q_curve $\mathbb{Q}$-curves modcurve.psl2index $\PSL_2$-index modcurve.sl2level $\SL_2$-level ring.a-field $A$-field modcurve.j_invariant_map $j$-invariant map specialfunction.k_bessel $K$-Bessel functions modcurve.gonality $K$-gonality of a modular curve nf.padic_completion $p$-adic completion of a number field lf.padic_field $p$-adic field group.pgroup $p$-group gg.tnumber $T$-number of a transitive group g2c.two_selmer_rank 2-Selmer rank g2c.two_torsion_field 2-torsion field