Documentation

Computer • algebra

Documentation

Introduction
- Modular Forms
- About the Package
- Categories
- Verbose Output
  - Example ModFrm_Basics (H141E1)
- An Illustrative Overview
  - Example ModFrm_Overview (H141E2)
Creation Functions
- Ambient Spaces
- Base Extension
- Elements
Bases
- Basis(M) : ModFrm -> SeqEnum
- Basis(M, prec) : ModFrm, RngIntElt -> [RngSerPowElt]
- PrecisionBound(M : parameters) : ModFrm -> RngIntElt
- RModule(M) : ModFrm -> ModTupRng, Map
- Example ModFrm_Bases (H141E6)
q-Expansions
- qExpansion(f) : ModFrmElt -> RngSerPowElt
- Coefficient(f, n) : ModFrmElt, RngIntElt -> RngElt
- Precision(M) : ModFrm -> RngIntElt
- Example ModFrm_qExpansion (H141E7)
- Example ModFrm_WeierstrassPoints (H141E8)
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 ModFrm_Arithmetic (H141E9)
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 ModFrm_Predicates (H141E10)
Properties
- AmbientSpace(M) : ModFrm -> ModFrm
- BaseRing(M) : ModFrm -> Rng
- Degree(f) : ModFrmElt -> RngIntElt
- Dimension(M) : ModFrm -> RngIntElt
- DimensionByFormula(M) : ModFrm -> RngIntElt
- DimensionByFormula(N, k) : RngIntElt, 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 ModFrm_Properties (H141E11)
Subspaces
- ZeroSubspace(M) : ModFrm -> ModFrm
- CuspidalSubspace(M) : ModFrm -> ModFrm
- EisensteinSubspace(M) : ModFrm -> ModFrm
- EisensteinProjection(f) : ModFrmElt -> ModFrmElt
- NewSubspace(M) : ModFrm -> ModFrm
- DihedralSubspace(M) : ModFrm -> ModFrm
- Example ModFrm_Subspaces (H141E12)
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 ModFrm_HeckePolynomials (H141E13)
Eisenstein Series
- EisensteinSeries(M) : ModFrm -> List
- IsEisensteinSeries(f) : ModFrmElt -> BoolElt
- EisensteinData(f) : ModFrmElt -> Tup
- Example ModFrm_EisensteinSeries (H141E14)
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
- Example ModFrm_Newforms (H141E15)
- Labels
  - Newforms(label) : MonStgElt -> ModFrmElt
  - Example ModFrm_NewformLabeling (H141E16)
Reductions and Embeddings
- Reductions(f, p) : ModFrmElt, RngIntElt -> List
- pAdicEmbeddings(f, p) : ModFrmElt, RngIntElt -> List
- ComplexEmbeddings(f) : ModFrmElt -> List
- Example ModFrm_ReductionsAndEmbeddings (H141E17)
Congruences
- CongruenceGroup(M1, M2, prec) : ModFrm, ModFrm, RngIntElt -> GrpAb
- CongruenceGroupAnemic(M1, M2, prec) : ModFrm, ModFrm, RngIntElt -> GrpAb
- Example ModFrm_Congruences (H141E18)
Overconvergent Modular Forms
- OverconvergentHeckeSeriesDegreeBound(p, N, k, m) : RngIntElt, RngIntElt, RngIntElt, RngIntElt -> RngIntElt
- OverconvergentHeckeSeries(p, N, k, m) : RngIntElt, RngIntElt, RngIntElt, RngIntElt -> RngUPolElt
- Example ModFrm_overconvergent (H141E19)
Algebraic Relations
- Relations(M, d, prec) : ModFrm, RngIntElt, RngIntElt -> SeqEnum
- Example ModFrm_Relations (H141E20)
Elliptic Curves
- ModularForm(E) : CrvEll -> ModFrmElt
- Eigenform(E, prec) : CrvEll, RngIntElt -> RngSerPowElt
- EllipticCurve(f) : ModFrmElt -> CrvEll
- Example ModFrm_EllipticCurves (H141E21)
Modular Symbols
- ModularSymbols(M) : ModFrm -> SeqEnum
- ModularSymbols(M, sign) : ModFrm, RngIntElt -> ModSym
- Example ModFrm_ModularSymbols (H141E22)
Bibliography

V2.28, 13 July 2023

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