# Modular Symbols

- [Introduction](introduction.md)

  - [Modular Symbols](introduction.md#modular-symbols)

- [Basics](introduction-2.md)

  - [Verbose Output](introduction-2.md#verbose-output)

  - [Categories](introduction-2.md#categories)

    - [`Example: Creation`](introduction-2.md#example-ex-dfabe0)

- [Creation Functions](creation.md)

  - [Ambient Spaces](creation.md#ambient-spaces)

    - [`ModularSymbols(N): RngIntElt → ModSym`](creation.md#function-modularsymbols-rngintelt)

    - [`ModularSymbols(N, k): RngIntElt, RngIntElt → ModSym`](creation.md#function-modularsymbols-rngintelt-rngintelt)

    - [`ModularSymbols(N, k, F): RngIntElt, RngIntElt, Fld → ModSym`](creation.md#function-modularsymbols-rngintelt-rngintelt-fld)

    - [`ModularSymbols(N, k, sign): RngIntElt, RngIntElt, RngIntElt → ModSym`](creation.md#function-modularsymbols-rngintelt-rngintelt-rngintelt)

    - [`ModularSymbols(N, k, F, sign): RngIntElt, RngIntElt, Fld, RngIntElt → ModSym`](creation.md#function-modularsymbols-rngintelt-rngintelt-fld-rngintelt)

    - [`ModularSymbols(eps, k): GrpDrchElt, RngIntElt → ModSym`](creation.md#function-modularsymbols-grpdrchelt-rngintelt)

    - [`ModularSymbols(eps, k, sign): GrpDrchElt, RngIntElt, RngIntElt → ModSym`](creation.md#function-modularsymbols-grpdrchelt-rngintelt-rngintelt)

    - [`Example: Creation Ambient`](creation.md#example-ex-3cf3aa)

  - [Labels](creation.md#labels)

    - [`ModularSymbols(s, sign): MonStgElt, RngIntElt → ModSym`](creation.md#function-modularsymbols-monstgelt-rngintelt)

    - [`ModularSymbols(s): MonStgElt → ModSym`](creation.md#function-modularsymbols-monstgelt)

    - [`Example: Creation Spaces`](creation.md#example-ex-999030)

  - [Creation of Elements](creation.md#creation-of-elements)

    - [`Example: Creation Elements 1`](creation.md#example-ex-c14194)

    - [`Example: Representation Conversion`](creation.md#example-ex-b624a4)

    - [`M ! x: ModSym, . → ModSymElt`](creation.md#operation-op-modsym)

    - [`ConvertFromManinSymbol(M, x): ModSym, Tup → ModSymElt`](creation.md#function-convertfrommaninsymbol-modsym-tup)

    - [`ManinSymbol(x): ModSymElt → SeqEnum`](creation.md#function-maninsymbol-modsymelt)

    - [`Example: Creation Elements 2`](creation.md#example-ex-4a3707)

- [Bases](bases.md)

  - [`Basis(M): ModSym → SeqEnum`](bases.md#function-basis-modsym)

  - [`IntegralBasis(M): ModSym → SeqEnum`](bases.md#function-integralbasis-modsym)

  - [`Example: Integral Basis`](bases.md#example-ex-e6fd22)

- [Associated Vector Space](underlying-representation.md)

  - [`VectorSpace(M): ModSym → ModTupFld, Map, Map`](underlying-representation.md#function-vectorspace-modsym)

  - [`DualVectorSpace(M): ModSym → ModTupFld`](underlying-representation.md#function-dualvectorspace-modsym)

  - [`Lattice(M): ModSym → Lat`](underlying-representation.md#function-lattice-modsym)

  - [`Example: Representation`](underlying-representation.md#example-ex-fb3783)

- [Degeneracy Maps](changing-level.md)

  - [`DegeneracyMap(M1, M2, d): ModSym, ModSym, RngIntElt → Map`](changing-level.md#function-degeneracymap-modsym-modsym-rngintelt)

  - [`DegeneracyMatrix(M1, M2, d): ModSym, ModSym, RngIntElt → AlgMatElt`](changing-level.md#function-degeneracymatrix-modsym-modsym-rngintelt)

  - [`ModularSymbols(M, N'): ModSym, RngIntElt → ModSym`](changing-level.md#function-modularsymbols-modsym-rngintelt)

  - [`M1 !! M2: ModSym, ModSym → ModSym`](changing-level.md#operation-op-modsym-modsym)

  - [`Example: Coercion Spaces`](changing-level.md#example-ex-9b0555)

- [Decomposition](decomposition.md)

  - [`Decomposition(M, bound : parameters): ModSym, RngIntElt → SeqEnum`](decomposition.md#function-decomposition-modsym-rngintelt)

  - [`NewformDecomposition(M : parameters): ModSym → SeqEnum`](decomposition.md#function-newformdecomposition-modsym)

  - [`AssociatedNewSpace(M): ModSym → ModSym`](decomposition.md#function-associatednewspace-modsym)

  - [`SortDecomposition(D): [ModSym] → SeqEnum`](decomposition.md#function-sortdecomposition-modsym)

  - [`IsIrreducible(M): ModSym → BoolElt`](decomposition.md#function-isirreducible-modsym)

  - [`M1 lt M2: ModSym, ModSym → BoolElt`](decomposition.md#operation-op-lt-modsym-modsym)

  - [`Example: Decomposition`](decomposition.md#example-ex-25e8b4)

- [Subspaces](subspaces.md)

  - [`CuspidalSubspace(M): ModSym → ModSym`](subspaces.md#function-cuspidalsubspace-modsym)

  - [`IsCuspidal(M): ModSym → BoolElt`](subspaces.md#function-iscuspidal-modsym)

  - [`EisensteinSubspace(M): ModSym → ModSym`](subspaces.md#function-eisensteinsubspace-modsym)

  - [`IsEisenstein(M): ModSym → BoolElt`](subspaces.md#function-iseisenstein-modsym)

  - [`NewSubspace(M): ModSym → ModSym`](subspaces.md#function-newsubspace-modsym)

  - [`IsNew(M): ModSym → BoolElt`](subspaces.md#function-isnew-modsym)

  - [`NewSubspace(M, p): ModSym, RngIntElt → ModSym`](subspaces.md#function-newsubspace-modsym-rngintelt)

  - [`Kernel(I, M): [Tup], ModSym → ModSym`](subspaces.md#function-kernel-tup-modsym)

  - [`Complement(M): ModSym → ModSym`](subspaces.md#function-complement-modsym)

  - [`BoundaryMap(M): ModSym → ModMatFldElt`](subspaces.md#function-boundarymap-modsym)

  - [`Example: Subspaces`](subspaces.md#example-ex-a39393)

  - [`Example: Boundary Map`](subspaces.md#example-ex-2e2403)

- [Twists](twists.md)

  - [`IsTwist(M1, M2, p): ModSym, ModSym, RngIntElt → BoolElt, GrpDrchElt`](twists.md#function-istwist-modsym-modsym-rngintelt)

  - [`IsMinimalTwist(M, p): ModSym, RngIntElt → BoolElt, ModSym, GrpDrchElt`](twists.md#function-isminimaltwist-modsym-rngintelt)

  - [`Example Twists`](twists.md#example-ex-1f6206)

- [Operators](operators.md)

  - [`Example: Hecke Operators`](operators.md#example-ex-223884)

  - [`HeckeOperator(M, n): ModSym, RngIntElt → AlgMatElt`](operators.md#function-heckeoperator-modsym-rngintelt)

  - [`HeckePolynomial(M, n): ModSym, RngIntElt → RngUPolResElt`](operators.md#function-heckepolynomial-modsym-rngintelt)

  - [`IntegralHeckeOperator(M, n): ModSym, RngIntElt → AlgMatElt`](operators.md#function-integralheckeoperator-modsym-rngintelt)

  - [`DualHeckeOperator(M, n): ModSym, RngIntElt → AlgMatElt`](operators.md#function-dualheckeoperator-modsym-rngintelt)

  - [`AtkinLehner(M, q): ModSym, RngIntElt → AlgMatElt`](operators.md#function-atkinlehner-modsym-rngintelt)

  - [`DualAtkinLehner(M, q): ModSym, RngIntElt → AlgMatElt`](operators.md#function-dualatkinlehner-modsym-rngintelt)

  - [`StarInvolution(M): ModSym → AlgMatElt`](operators.md#function-starinvolution-modsym)

  - [`DualStarInvolution(M): ModSym → AlgMatElt`](operators.md#function-dualstarinvolution-modsym)

  - [`ThetaOperator(M1, M2): ModSym, ModSym → Map`](operators.md#function-thetaoperator-modsym-modsym)

  - [`Example: Operators`](operators.md#example-ex-e83c93)

  - [`Example: Theta Operator`](operators.md#example-ex-2cde37)

- [The Hecke Algebra](hecke-algebra.md)

  - [`HeckeBound(M): ModSym → RngIntElt`](hecke-algebra.md#function-heckebound-modsym)

  - [`SetHeckeBound(M, n): ModSym, RngIntElt → RngIntElt`](hecke-algebra.md#function-setheckebound-modsym-rngintelt)

  - [`HeckeAlgebra(M : Bound): ModSym → AlgMat`](hecke-algebra.md#function-heckealgebra-modsym)

  - [`DiscriminantOfHeckeAlgebra(M : Bound): ModSym → RngIntElt`](hecke-algebra.md#function-discriminantofheckealgebra-modsym)

  - [`HeckeEigenvalueRing(M : parameters): ModSym → Rng, Map`](hecke-algebra.md#function-heckeeigenvaluering-modsym)

  - [`HeckeEigenvalueField(M): ModSym → Fld, Map`](hecke-algebra.md#function-heckeeigenvaluefield-modsym)

  - [`Example: Hecke Algebra`](hecke-algebra.md#example-ex-1119e4)

- [The Intersection Pairing](intersection_pairing.md)

  - [`IntersectionPairing(x, y): ModSymElt, ModSymElt → FldRatElt`](intersection_pairing.md#function-intersectionpairing-modsymelt-modsymelt)

  - [`Example: Intersection Pairing`](intersection_pairing.md#example-ex-3889b3)

- [$q$-Expansions](qexpansions.md)

  - [`Eigenform(M, prec): ModSym, RngIntElt → RngSerPowElt`](qexpansions.md#function-eigenform-modsym-rngintelt)

  - [`Eigenform(M): ModSym → RngSerPowElt`](qexpansions.md#function-eigenform-modsym)

  - [`qEigenform(M, prec): ModSym, RngIntElt → RngSerPowElt`](qexpansions.md#function-qeigenform-modsym-rngintelt)

  - [`qEigenform(M): ModSym → RngSerPowElt`](qexpansions.md#function-qeigenform-modsym)

  - [`PowerSeries(M, prec): ModSym, RngIntElt → RngSerPowElt`](qexpansions.md#function-powerseries-modsym-rngintelt)

  - [`PowerSeries(M): ModSym → RngSerPowElt`](qexpansions.md#function-powerseries-modsym)

  - [`qExpansionBasis(M, prec : parameters): ModSym, RngIntElt → SeqEnum`](qexpansions.md#function-qexpansionbasis-modsym-rngintelt)

  - [`qIntegralBasis(M): ModSym → SeqEnum`](qexpansions.md#function-qintegralbasis-modsym)

  - [`qIntegralBasis(M, prec): ModSym, RngIntElt → SeqEnum`](qexpansions.md#function-qintegralbasis-modsym-rngintelt)

  - [`qIntegralBasis(seq, prec): [ModSym], RngIntElt → SeqEnum`](qexpansions.md#function-qintegralbasis-modsym-rngintelt-2)

  - [`SystemOfEigenvalues(M, prec): ModSym, RngIntElt → SeqEnum`](qexpansions.md#function-systemofeigenvalues-modsym-rngintelt)

  - [`Example: q Expansions`](qexpansions.md#example-ex-cb0791)

- [Special Values of $L$-functions](l-series.md)

  - [`LSeries(M, j, prec): ModSym, RngIntElt, RngIntElt → FldPrElt`](l-series.md#function-lseries-modsym-rngintelt-rngintelt)

  - [`LSeriesLeadingCoefficient(M, j, prec): ModSym, RngIntElt, RngIntElt → FldPrElt, RngIntElt`](l-series.md#function-lseriesleadingcoefficient-modsym-rngintelt-rngintelt)

  - [`RealVolume(M, prec): ModSym, RngIntElt → FldPrElt`](l-series.md#function-realvolume-modsym-rngintelt)

  - [`MinusVolume(M, prec): ModSym, RngIntElt → FldPrElt`](l-series.md#function-minusvolume-modsym-rngintelt)

  - [`LRatio(M, j : parameters): ModSym, RngIntElt → FldRatElt`](l-series.md#function-lratio-modsym-rngintelt)

  - [`LRatioOddPart(M, j): ModSym, RngIntElt → FldRatElt`](l-series.md#function-lratiooddpart-modsym-rngintelt)

  - [`Example: L Series`](l-series.md#example-ex-75fe6d)

  - [Winding Elements](l-series.md#winding-elements)

    - [`WindingElement(M): ModSym → ModSymElt`](l-series.md#function-windingelement-modsym)

    - [`WindingElement(M, i): ModSym, RngIntElt → ModSymElt`](l-series.md#function-windingelement-modsym-rngintelt)

    - [`TwistedWindingElement(M, i, eps): ModSym, RngIntElt, GrpDrchElt → ModSymElt`](l-series.md#function-twistedwindingelement-modsym-rngintelt-grpdrchelt)

    - [`WindingLattice(M, j : parameters): ModSym, RngIntElt → Lat`](l-series.md#function-windinglattice-modsym-rngintelt)

    - [`WindingSubmodule(M, j : parameters): ModSym, RngIntElt → ModTupFld`](l-series.md#function-windingsubmodule-modsym-rngintelt)

    - [`TwistedWindingSubmodule(M, j, eps): ModSym, RngIntElt, GrpDrchElt → ModTupFld`](l-series.md#function-twistedwindingsubmodule-modsym-rngintelt-grpdrchelt)

- [The Associated Complex Torus](complex-tori.md)

  - [`SubgroupOfTorus(M, x): ModSym, ModSymElt → RngIntElt`](complex-tori.md#function-subgroupoftorus-modsym-modsymelt)

  - [`SubgroupOfTorus(M, s): ModSym, SeqEnum → GrpAb`](complex-tori.md#function-subgroupoftorus-modsym-seqenum)

  - [`Example: Cuspidal Subgroup`](complex-tori.md#example-ex-7942de)

  - [`Example: Cuspidal Subgroup Table`](complex-tori.md#example-ex-cbaa0c)

  - [`ModularKernel(M): ModSym → GrpAb`](complex-tori.md#function-modularkernel-modsym)

  - [`CongruenceGroup(M : parameters): ModSym → GrpAb`](complex-tori.md#function-congruencegroup-modsym)

  - [`IntersectionGroup(M1, M2): ModSym, ModSym → GrpAb`](complex-tori.md#function-intersectiongroup-modsym-modsym)

  - [`IntersectionGroup(S): SeqEnum → GrpAb`](complex-tori.md#function-intersectiongroup-seqenum)

  - [`Example: BSD`](complex-tori.md#example-ex-a9314d)

  - [The Period Map](complex-tori.md#the-period-map)

    - [`PeriodMapping(M, n): ModSym, RngIntElt → Map`](complex-tori.md#function-periodmapping-modsym-rngintelt)

    - [`PeriodMapping(M): ModSym → Map`](complex-tori.md#function-periodmapping-modsym)

    - [`Periods(M, prec): ModSym, RngIntElt → SeqEnum`](complex-tori.md#function-periods-modsym-rngintelt)

    - [`Periods(M): ModSym → SeqEnum`](complex-tori.md#function-periods-modsym)

    - [`ClassicalPeriod(M, j, prec): ModSym, RngIntElt, RngIntElt → FldPrElt`](complex-tori.md#function-classicalperiod-modsym-rngintelt-rngintelt)

  - [Projection Mappings](complex-tori.md#projection-mappings)

    - [`RationalMapping(M): ModSym → Map`](complex-tori.md#function-rationalmapping-modsym)

    - [`IntegralMapping(M): ModSym → Map`](complex-tori.md#function-integralmapping-modsym)

    - [`Example: Modular Ab Var Rational`](complex-tori.md#example-ex-a62806)

- [Modular Abelian Varieties](modular-abvars.md)

  - [Modular Degree and Torsion](modular-abvars.md#modular-degree-and-torsion)

    - [`ModularDegree(M): ModSym → RngIntElt`](modular-abvars.md#function-modulardegree-modsym)

    - [`CongruenceModulus(M : parameters): ModSym → RngIntElt`](modular-abvars.md#function-congruencemodulus-modsym)

    - [`TorsionBound(M, maxp): ModSym, RngIntElt → RngIntElt`](modular-abvars.md#function-torsionbound-modsym-rngintelt)

    - [`Example: Modular Ab Var Arithmetic`](modular-abvars.md#example-ex-9fc9de)

  - [Tamagawa Numbers and Orders of Component Groups](modular-abvars.md#tamagawa-numbers-and-orders-of-component-groups)

    - [`ComponentGroupOrder(M, p): ModSym, RngIntElt → RngIntElt`](modular-abvars.md#function-componentgrouporder-modsym-rngintelt)

    - [`TamagawaNumber(M, p): ModSym, RngIntElt → RngIntElt`](modular-abvars.md#function-tamagawanumber-modsym-rngintelt)

    - [`RealTamagawaNumber(M): ModSym → RngIntElt`](modular-abvars.md#function-realtamagawanumber-modsym)

    - [`MinusTamagawaNumber(M): ModSym → RngIntElt`](modular-abvars.md#function-minustamagawanumber-modsym)

    - [`Example: Modular Ab Var Comp Grp`](modular-abvars.md#example-ex-402be9)

- [Elliptic Curves](elliptic-curves.md)

  - [`ModularSymbols(E): CrvEll → ModSym`](elliptic-curves.md#function-modularsymbols-crvell)

  - [`ModularSymbols(E, sign): CrvEll, RngIntElt → ModSym`](elliptic-curves.md#function-modularsymbols-crvell-rngintelt)

  - [`EllipticCurve(M): ModSym → CrvEll`](elliptic-curves.md#function-ellipticcurve-modsym)

  - [`pAdicLSeries(E, p): CrvEll, RngIntElt → RngSerPowElt`](elliptic-curves.md#function-padiclseries-crvell-rngintelt)

  - [`Example: BSD389A`](elliptic-curves.md#example-ex-7194fd)

- [Dimension Formulas](dimension-formulas.md)

  - [`DimensionCuspFormsGamma0(N, k): RngIntElt, RngIntElt → RngIntElt`](dimension-formulas.md#function-dimensioncuspformsgamma0-rngintelt-rngintelt)

  - [`DimensionNewCuspFormsGamma0(N, k): RngIntElt, RngIntElt → RngIntElt`](dimension-formulas.md#function-dimensionnewcuspformsgamma0-rngintelt-rngintelt)

  - [`DimensionCuspFormsGamma1(N, k): RngIntElt, RngIntElt → RngIntElt`](dimension-formulas.md#function-dimensioncuspformsgamma1-rngintelt-rngintelt)

  - [`DimensionNewCuspFormsGamma1(N, k): RngIntElt, RngIntElt → RngIntElt`](dimension-formulas.md#function-dimensionnewcuspformsgamma1-rngintelt-rngintelt)

  - [`DimensionCuspForms(eps, k): GrpDrchElt, RngIntElt → RngIntElt`](dimension-formulas.md#function-dimensioncuspforms-grpdrchelt-rngintelt)

  - [`Example: Dimension Formulas`](dimension-formulas.md#example-ex-57206a)