# Hilbert Modular Forms

- [Introduction](introduction.md)

  - [Definitions and Background](introduction.md#definitions-and-background)

  - [Algorithms and the Jacquet-Langlands Correspondence](introduction.md#algorithms-and-the-jacquet-langlands-correspondence)

  - [Algorithm I (Using Definite Quaternion Orders)](introduction.md#algorithm-i-using-definite-quaternion-orders)

  - [Algorithm II (Using Indefinite Quaternion Orders)](introduction.md#algorithm-ii-using-indefinite-quaternion-orders)

  - [Categories](introduction.md#categories)

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

- [Creation of Full Cuspidal Spaces](creation.md)

  - [`HilbertCuspForms(F, N, k): FldNum, RngOrdIdl, SeqEnum → ModFrmHil`](creation.md#function-hilbertcuspforms-fldnum-rngordidl-seqenum)

  - [`HilbertCuspForms(F, N): FldNum, RngOrdIdl → ModFrmHil`](creation.md#function-hilbertcuspforms-fldnum-rngordidl)

  - [`Example: Creation Example`](creation.md#example-ex-ae5c18)

- [Caching Spaces of Modular Forms](hmf-caching.md)

  - [`SetStoreModularForms(F, v): Fld, BoolElt`](hmf-caching.md#function-setstoremodularforms-fld-boolelt)

  - [`ClearStoredModularForms(F): Fld`](hmf-caching.md#function-clearstoredmodularforms-fld)

- [Basic Properties](basic-properties.md)

  - [`BaseField(M): ModFrmHil`](basic-properties.md#function-basefield-modfrmhil)

  - [`BaseRing(M): ModFrmHil`](basic-properties.md#function-basering-modfrmhil)

  - [`CoefficientField(M): ModFrmHil`](basic-properties.md#function-coefficientfield-modfrmhil)

  - [`CoefficientRing(M): ModFrmHil`](basic-properties.md#function-coefficientring-modfrmhil)

  - [`Weight(M): ModFrmHil → SeqEnum[RngIntElt]`](basic-properties.md#function-weight-modfrmhil)

  - [`CentralCharacter(M): ModFrmHil → RngIntElt`](basic-properties.md#function-centralcharacter-modfrmhil)

  - [`Level(M): ModFrmHil → RngOrdIdl`](basic-properties.md#function-level-modfrmhil)

  - [`DirichletCharacter(M): ModFrmHil → GrpDrchNFElt`](basic-properties.md#function-dirichletcharacter-modfrmhil)

  - [`IsCuspidal(M): ModFrmHil → BoolElt`](basic-properties.md#function-iscuspidal-modfrmhil)

  - [`IsNew(M): ModFrmHil → BoolElt`](basic-properties.md#function-isnew-modfrmhil)

  - [`NewLevel(M): ModFrmHil → RngOrdIdl`](basic-properties.md#function-newlevel-modfrmhil)

  - [`Dimension(M): ModFrmHil → RngIntElt`](basic-properties.md#function-dimension-modfrmhil)

  - [`QuaternionOrder(M): ModFrmHil → AlgAssVOrd`](basic-properties.md#function-quaternionorder-modfrmhil)

  - [`IsDefinite(M): ModFrmHil → BoolElt`](basic-properties.md#function-isdefinite-modfrmhil)

  - [`Example: Basic Example`](basic-properties.md#example-ex-0a8a2f)

- [Elements](elements.md)

  - [`Parent(f): ModFrmHilElt → ModFrmHil`](elements.md#function-parent-modfrmhilelt)

  - [`BaseField(f): ModFrmHilElt → Fld`](elements.md#function-basefield-modfrmhilelt)

- [Operators](operators.md)

  - [`HeckeOperator(M, P): ModFrmHil, RngOrdIdl → Mtrx`](operators.md#function-heckeoperator-modfrmhil-rngordidl)

  - [`Example: hecke`](operators.md#example-ex-e87578)

  - [`AtkinLehnerOperator(M, P): ModFrmHil, RngOrdIdl → Mtrx`](operators.md#function-atkinlehneroperator-modfrmhil-rngordidl)

  - [`DegeneracyOperator(M, P, Q): ModFrmHil, RngOrdIdl, RngOrdIdl → Mtrx`](operators.md#function-degeneracyoperator-modfrmhil-rngordidl-rngordidl)

  - [`DeleteHeckePrecomputation(O): AlgAssVOrd`](operators.md#function-deleteheckeprecomputation-algassvord)

  - [`DeleteHeckePrecomputation(O, P): AlgAssVOrd, RngOrdIdl`](operators.md#function-deleteheckeprecomputation-algassvord-rngordidl)

- [Creation of Subspaces](subspaces.md)

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

  - [`NewSubspace(M, I): ModFrmHil, RngOrdIdl → ModFrmHil`](subspaces.md#function-newsubspace-modfrmhil-rngordidl)

  - [`SetRationalBasis(M): ModFrmHil`](subspaces.md#function-setrationalbasis-modfrmhil)

  - [`Example: Newsubspace Example1`](subspaces.md#example-ex-863bee)

  - [`Example: Newsubspace Example2`](subspaces.md#example-ex-375676)

- [Eigenspace Decomposition and Eigenforms](eigenforms.md)

  - [`HeckeEigenvalueBound(M, P): ModFrmHil, RngOrdIdl → RngIntElt`](eigenforms.md#function-heckeeigenvaluebound-modfrmhil-rngordidl)

  - [`NewformDecomposition(M): ModFrmHil → List`](eigenforms.md#function-newformdecomposition-modfrmhil)

  - [`NewformsOfDegree1(M): ModFrmHil → List`](eigenforms.md#function-newformsofdegree1-modfrmhil)

  - [`Eigenform(M): ModFrmHil → ModFrmHilElt`](eigenforms.md#function-eigenform-modfrmhil)

  - [`Eigenforms(M): ModFrmHil → List`](eigenforms.md#function-eigenforms-modfrmhil)

  - [`HeckeEigenvalueField(M): ModFrmHil → Fld`](eigenforms.md#function-heckeeigenvaluefield-modfrmhil)

  - [`HeckeEigenvalue(f, P): ModFrmHilElt, RngOrdIdl → FldAlgElt`](eigenforms.md#function-heckeeigenvalue-modfrmhilelt-rngordidl)

  - [`Example: Eigenform Examples`](eigenforms.md#example-ex-15f603)

- [Further Examples](further-examples.md)

  - [`Example: Integral Hecke`](further-examples.md#example-ex-3b8963)

  - [`Example: Classical Example`](further-examples.md#example-ex-c19462)