Modular Forms

• Introduction
  • Modular Forms
  • About the Package
  • Categories
  • Verbose Output
    • Example: Basics
  • An Illustrative Overview
    • Example: Overview
• Creation Functions
  • Ambient Spaces
    • ModularForms(N): RngIntElt → ModFrm
    • ModularForms(N, k): RngIntElt, RngIntElt → ModFrm
    • ModularForms(eps, k): GrpDrchElt, RngIntElt → ModFrm
    • ModularForms(chars, k): [GrpDrchElt], RngIntElt → ModFrm
    • ModularForms(G, k): GrpPSL2, RngIntElt → ModFrm
    • ModularForms(G): GrpPSL2 → ModFrm
    • CuspForms(x): Any → ModFrm
    • CuspForms(x, y): Any, Any → ModFrm
    • Example: Creation Space
    • Half-integral Weight Forms
      • HalfIntegralWeightForms(N, w): RngIntElt, FldRatElt → ModFrm
      • HalfIntegralWeightForms(chi, w): GrpDrchElt, FldRatElt → ModFrm
      • HalfIntegralWeightForms(G, w): GrpPSL2, FldRatElt → ModFrm
  • Base Extension
    • BaseExtend(M, R): ModFrm, Rng → ModFrm, Map
    • BaseExtend(M, phi): ModFrm, Map → ModFrm, Map
    • Example: Base Extend
  • Elements
    • M . i: ModFrm, RngIntElt → ModFrmElt
    • M ! f: ModFrm, . → ModFrmElt
    • ModularForm(E): CrvEll → ModFrm
    • Example: Creation Elements
• Bases
  • Basis(M): ModFrm → SeqEnum
  • Basis(M, prec): ModFrm, RngIntElt → [RngSerPowElt]
  • qExpansionBasis(M, prec): ModFrm, RngIntElt → [RngSerPowElt]
  • PrecisionBound(M : parameters): ModFrm → RngIntElt
  • RModule(M): ModFrm → ModTupRng, Map
  • RSpace(M): ModFrm → ModTupRng, Map
  • VectorSpace(M): ModFrm → ModTupRng, Map
  • Example: Bases
• \(q\)-Expansions
  • qExpansion(f): ModFrmElt → RngSerPowElt
  • qExpansion(f, prec): ModFrmElt, RngIntElt → RngSerPowElt
  • PowerSeries(f): ModFrmElt → RngSerPowElt
  • PowerSeries(f, prec): ModFrmElt, RngIntElt → RngSerPowElt
  • Coefficient(f, n): ModFrmElt, RngIntElt → RngElt
  • Precision(M): ModFrm → RngIntElt
  • SetPrecision(M, prec): ModFrm, RngIntElt
  • Example: q Expansion
  • Example: Weierstrass Points
• Arithmetic
  • f + g: ModFrmElt, ModFrmElt → ModFrmElt
  • f + g: ModFrmElt, RngSerPowElt → RngSerPowElt
  • f - g: ModFrmElt, ModFrmElt → ModFrmElt
  • a * f: RngElt, ModFrmElt → ModFrmElt
  • f / a: ModFrmElt, RngElt → ModFrmElt
  • f ^ n: ModFrmElt, RngIntElt → ModFrmElt
  • f * g: ModFrmElt, ModFrmElt → ModFrmElt
  • Example: Arithmetic
• Predicates
  • IsAmbientSpace(M): ModFrm → BoolElt
  • IsCuspidal(M): ModFrm → BoolElt
  • IsEisenstein(M): ModFrm → BoolElt
  • IsEisensteinSeries(f): ModFrmElt → BoolElt
  • IsGamma0(M): ModFrm → BoolElt
  • IsGamma1(M): ModFrm → BoolElt
  • IsNew(M): ModFrm → BoolElt
  • IsNewform(f): ModFrmElt → BoolElt
  • IsRingOfAllModularForms(M): ModFrm → BoolElt
  • Example: Predicates
• Properties
  • AmbientSpace(M): ModFrm → ModFrm
  • BaseRing(M): ModFrm → Rng
  • CoefficientRing(M): ModFrm → Rng
  • Degree(f): ModFrmElt → RngIntElt
  • Dimension(M): ModFrm → RngIntElt
  • DimensionByFormula(M): ModFrm → RngIntElt
  • DimensionByFormula(N, k): RngIntElt, FldRatElt → RngIntElt
  • DimensionByFormula(chi, k): GrpDrchElt, FldRatElt → RngIntElt
  • DimensionByFormula(N, chi, k): RngIntElt, GrpDrchElt, FldRatElt → RngIntElt
  • DirichletCharacters(M): ModFrm → [GrpDrchElt]
  • DirichletCharacter(f): ModFrmElt → GrpDrchElt
  • Eltseq(f): ModFrmElt → SeqEnum
  • Level(f): ModFrmElt → RngIntElt
  • Level(M): ModFrm → RngIntElt
  • Weight(f): ModFrmElt → RngIntElt
  • Weight(M): ModFrm → RngIntElt
  • WeightOneHalfData(H): ModFrm → List
  • Example: Properties
• Subspaces
  • ZeroSubspace(M): ModFrm → ModFrm
  • CuspidalSubspace(M): ModFrm → ModFrm
  • EisensteinSubspace(M): ModFrm → ModFrm
  • EisensteinProjection(f): ModFrmElt → ModFrmElt
  • CuspidalProjection(f): ModFrmElt → ModFrmElt
  • NewSubspace(M): ModFrm → ModFrm
  • DihedralSubspace(M): ModFrm → ModFrm
  • Example: Subspaces
• Operators
  • HeckeOperator(M, n): ModFrm, RngIntElt → AlgMatElt
  • HeckeOperator(n,f): RngIntElt, ModFrmElt → ModFrmElt
  • HeckePolynomial(M, n : parameters): ModFrm, RngIntElt → RngUPolElt
  • AtkinLehnerOperator(M, q): ModFrm, RngIntElt → AlgMatElt
  • AtkinLehnerOperator(q, f): RngIntElt, ModFrmElt → ModFrmElt
  • Example: Hecke Polynomials
• Eisenstein Series
  • EisensteinSeries(M): ModFrm → List
  • IsEisensteinSeries(f): ModFrmElt → BoolElt
  • EisensteinData(f): ModFrmElt → Tup
  • Example: Eisenstein Series
• Weight Half Forms
  • WeightOneHalfData(M): ModFrm → List
• Weight One Forms
  • DihedralForms(M): ModFrm → List
• Newforms
  • NumberOfNewformClasses(M : parameters): ModFrm → RngIntElt
  • Newform(M, i, j : parameters): ModFrm, RngIntElt, RngIntElt → ModFrmElt
  • Newform(M, i : parameters): ModFrm, RngIntElt → ModFrmElt
  • Newforms(M : parameters): ModFrm → List
  • Newforms(I, M): [Tup], ModFrm → ModFrm
  • NewformDecomposition(M): ModFrm → List
  • Example: Newforms
  • Labels
    • Newforms(label): MonStgElt → ModFrmElt
    • Example: Newform Labeling
• Reductions and Embeddings
  • Reductions(f, p): ModFrmElt, RngIntElt → List
  • pAdicEmbeddings(f, p): ModFrmElt, RngIntElt → List
  • ComplexEmbeddings(f): ModFrmElt → List
  • Example: Reductions And Embeddings
• Congruences
  • CongruenceGroup(M1, M2, prec): ModFrm, ModFrm, RngIntElt → GrpAb
  • CongruenceGroupAnemic(M1, M2, prec): ModFrm, ModFrm, RngIntElt → GrpAb
  • Example: Congruences
• Overconvergent Modular Forms
  • OverconvergentHeckeSeriesDegreeBound(p, N, k, m): RngIntElt, RngIntElt, RngIntElt, RngIntElt → RngIntElt
  • OverconvergentHeckeSeries(p, N, k, m): RngIntElt, RngIntElt, RngIntElt, RngIntElt → RngUPolElt
  • OverconvergentHeckeSeries(p, N, k, m): RngIntElt, RngIntElt, [ RngIntElt ], RngIntElt → [ RngUPolElt ]
  • OverconvergentHeckeSeries(p, chi, k, m): RngIntElt, GrpDrchElt, RngIntElt, RngIntElt → RngUPolElt, RngPadElt
  • OverconvergentHeckeSeries(p, chi, k, m): RngIntElt, GrpDrchElt, [ RngIntElt ], RngIntElt → [ RngUPolElt ], RngPadElt
  • Example: overconvergent
• Algebraic Relations
  • Relations(M, d, prec): ModFrm, RngIntElt, RngIntElt → SeqEnum
  • Example: Relations
• Elliptic Curves
  • ModularForm(E): CrvEll → ModFrmElt
  • Newform(E): CrvEll → ModFrmElt
  • Eigenform(E, prec): CrvEll, RngIntElt → RngSerPowElt
  • qEigenform(E, prec): CrvEll, RngIntElt → RngSerPowElt
  • EllipticCurve(f): ModFrmElt → CrvEll
  • Example: Elliptic Curves
• Modular Symbols
  • ModularSymbols(M): ModFrm → SeqEnum
  • ModularSymbols(M, sign): ModFrm, RngIntElt → ModSym
  • Example: Modular Symbols