# Brandt Modules - [Introduction](ModBrdt:introduction.md) - [Brandt Module Creation](ModBrdt:brandt-modules.md) - [`BrandtModule(D): RngIntElt → ModBrdt`](ModBrdt:brandt-modules.md#function-brandtmodule-rngintelt) - [`BrandtModule(D, m): RngIntElt, RngIntElt → ModBrdt`](ModBrdt:brandt-modules.md#function-brandtmodule-rngintelt-rngintelt) - [`BrandtModule(A): AlgQuatOrd → ModBrdt`](ModBrdt:brandt-modules.md#function-brandtmodule-algquatord) - [`BrandtModule(A, R): AlgQuatOrd, Rng → ModBrdt`](ModBrdt:brandt-modules.md#function-brandtmodule-algquatord-rng) - [`BaseExtend(M, R): ModBrdt, Rng → ModBrdt`](ModBrdt:brandt-modules.md#function-baseextend-modbrdt-rng) - [`BrandtModule(M, N): AlgQuatOrd, RngElt → ModBrdt`](ModBrdt:brandt-modules.md#function-brandtmodule-algquatord-rngelt) - [`Example: Mod Brdt:Constructors`](ModBrdt:brandt-modules.md#example-ex-800093) - [Creation of Elements](ModBrdt:brandt-modules.md#creation-of-elements) - [`M ! x: ModBrdt, . → ModBrdtElt`](ModBrdt:brandt-modules.md#operation-op-modbrdt) - [`M . i: ModBrdt, RngIntElt → ModBrdtElt`](ModBrdt:brandt-modules.md#operation-operation-modbrdt-rngintelt-modbrdtelt) - [Operations on Elements](ModBrdt:brandt-modules.md#operations-on-elements) - [`a * x: RngElt, ModBrdtElt → ModBrdtElt`](ModBrdt:brandt-modules.md#operation-op-times-rngelt-modbrdtelt) - [`x * a: ModBrdtElt, RngElt → ModBrdtElt`](ModBrdt:brandt-modules.md#operation-op-times-modbrdtelt-rngelt) - [`x * T: ModBrdtElt, AlgMatElt → ModBrdtElt`](ModBrdt:brandt-modules.md#operation-op-times-modbrdtelt-algmatelt) - [`x + y: ModBrdtElt, ModBrdtElt → ModBrdtElt`](ModBrdt:brandt-modules.md#operation-op-plus-modbrdtelt-modbrdtelt) - [`x - y: ModBrdtElt, ModBrdtElt → ModBrdtElt`](ModBrdt:brandt-modules.md#operation-op-minus-modbrdtelt-modbrdtelt) - [`x eq y: ModBrdtElt, ModBrdtElt → BoolElt`](ModBrdt:brandt-modules.md#operation-op-eq-modbrdtelt-modbrdtelt) - [`Eltseq(x): ModBrdtElt → SeqEnum`](ModBrdt:brandt-modules.md#function-eltseq-modbrdtelt) - [`InnerProduct(x, y): ModBrdtElt, ModBrdtElt → RngElt`](ModBrdt:brandt-modules.md#function-innerproduct-modbrdtelt-modbrdtelt) - [`Norm(x): ModBrdtElt → RngElt`](ModBrdt:brandt-modules.md#function-norm-modbrdtelt) - [Categories and Parent](ModBrdt:brandt-modules.md#categories-and-parent) - [`Category(M): ModBrdt → Cat`](ModBrdt:brandt-modules.md#function-category-modbrdt) - [`Type(M): ModBrdt → Cat`](ModBrdt:brandt-modules.md#function-type-modbrdt) - [`Category(x): ModBrdtElt → Cat`](ModBrdt:brandt-modules.md#function-category-modbrdtelt) - [`Type(x): ModBrdtElt → Cat`](ModBrdt:brandt-modules.md#function-type-modbrdtelt) - [`Parent(x): ModBrdtElt → ModBrdt`](ModBrdt:brandt-modules.md#function-parent-modbrdtelt) - [`x in M: ModBrdtElt, ModBrdt → BoolElt`](ModBrdt:brandt-modules.md#operation-op-in-modbrdtelt-modbrdt) - [Elementary Invariants](ModBrdt:brandt-modules.md#elementary-invariants) - [`Level(M): ModBrdt → RngIntElt`](ModBrdt:brandt-modules.md#function-level-modbrdt) - [`Discriminant(M): ModBrdt → RngIntElt`](ModBrdt:brandt-modules.md#function-discriminant-modbrdt) - [`Conductor(M): ModBrdt → RngIntElt`](ModBrdt:brandt-modules.md#function-conductor-modbrdt) - [`BaseRing(M): ModBrdt → Rng`](ModBrdt:brandt-modules.md#function-basering-modbrdt) - [`Basis(M): ModBrdt → SeqEnum`](ModBrdt:brandt-modules.md#function-basis-modbrdt) - [Associated Structures](ModBrdt:brandt-modules.md#associated-structures) - [`AmbientModule(M): ModBrdt → ModBrdt`](ModBrdt:brandt-modules.md#function-ambientmodule-modbrdt) - [`IsAmbient(M): ModBrdt → BoolElt`](ModBrdt:brandt-modules.md#function-isambient-modbrdt) - [`Dimension(M): ModBrdt → RngIntElt`](ModBrdt:brandt-modules.md#function-dimension-modbrdt) - [`Rank(M): ModBrdt → RngIntElt`](ModBrdt:brandt-modules.md#function-rank-modbrdt) - [`Degree(M): ModBrdt → RngIntElt`](ModBrdt:brandt-modules.md#function-degree-modbrdt) - [`GramMatrix(M): ModBrdt → AlgMatElt`](ModBrdt:brandt-modules.md#function-grammatrix-modbrdt) - [`InnerProductMatrix(M): ModBrdt → AlgMatElt`](ModBrdt:brandt-modules.md#function-innerproductmatrix-modbrdt) - [`Example: ModBrdt:Module Creation`](ModBrdt:brandt-modules.md#example-ex-3bb6f4) - [`Ideals(M): ModBrdt → []`](ModBrdt:brandt-modules.md#function-ideals-modbrdt) - [Verbose Output](ModBrdt:brandt-modules.md#verbose-output) - [`Example: ModBrdt:Verbose Output`](ModBrdt:brandt-modules.md#example-ex-15591a) - [Subspaces and Decomposition](ModBrdt:Subspaces.md) - [`EisensteinSubspace(M): ModBrdt → ModBrdt`](ModBrdt:Subspaces.md#function-eisensteinsubspace-modbrdt) - [`CuspidalSubspace(M): ModBrdt → ModBrdt`](ModBrdt:Subspaces.md#function-cuspidalsubspace-modbrdt) - [`OrthogonalComplement(M): ModBrdt → ModBrdt`](ModBrdt:Subspaces.md#function-orthogonalcomplement-modbrdt) - [`M meet N: ModBrdt, ModBrdt → ModBrdt`](ModBrdt:Subspaces.md#operation-op-meet-modbrdt-modbrdt) - [`Decomposition(M, B): ModBrdt, RngIntElt → [ModBrdt]`](ModBrdt:Subspaces.md#function-decomposition-modbrdt-rngintelt) - [`SortDecomposition(D): [ModBrdt] → SeqEnum`](ModBrdt:Subspaces.md#function-sortdecomposition-modbrdt) - [`Example: Mod Brdt:Decomposition`](ModBrdt:Subspaces.md#example-ex-5a913e) - [Boolean Tests on Subspaces](ModBrdt:Subspaces.md#boolean-tests-on-subspaces) - [`IsEisenstein(M): ModBrdt → BoolElt`](ModBrdt:Subspaces.md#function-iseisenstein-modbrdt) - [`IsCuspidal(M): ModBrdt → BoolElt`](ModBrdt:Subspaces.md#function-iscuspidal-modbrdt) - [`IsIndecomposable(M, B): ModBrdt, RngIntElt → BoolElt`](ModBrdt:Subspaces.md#function-isindecomposable-modbrdt-rngintelt) - [`M1 subset M2: ModBrdt, ModBrdt → BoolElt`](ModBrdt:Subspaces.md#operation-op-subset-modbrdt-modbrdt) - [`M1 lt M2: ModBrdt, ModBrdt → BoolElt`](ModBrdt:Subspaces.md#operation-op-lt-modbrdt-modbrdt) - [`M1 gt M2: ModBrdt, ModBrdt → BoolElt`](ModBrdt:Subspaces.md#operation-op-gt-modbrdt-modbrdt) - [`Example: Mod Brdt:Eisenstein Subspace`](ModBrdt:Subspaces.md#example-ex-577acc) - [Hecke Operators](hecke-operators.md) - [`HeckeOperator(M, n): ModBrdt, RngIntElt → AlgMatElt`](hecke-operators.md#function-heckeoperator-modbrdt-rngintelt) - [`AtkinLehnerOperator(M, p): ModBrdt, RngIntElt → AlgMatElt`](hecke-operators.md#function-atkinlehneroperator-modbrdt-rngintelt) - [$q$-Expansions](qexpansions.md) - [`ThetaSeries(x, y, prec): ModBrdtElt, ModBrdtElt, RngIntElt → RngSerElt`](qexpansions.md#function-thetaseries-modbrdtelt-modbrdtelt-rngintelt) - [`qExpansionBasis(M, prec): ModBrdt, RngIntElt → SeqEnum`](qexpansions.md#function-qexpansionbasis-modbrdt-rngintelt) - [Dimensions of Spaces](ModBrdt:dimension-formulas.md) - [`BrandtModuleDimension(D, N): RngIntElt, RngIntElt → RngIntElt`](ModBrdt:dimension-formulas.md#function-brandtmoduledimension-rngintelt-rngintelt) - [`Example: Mod Brdt:Dimension`](ModBrdt:dimension-formulas.md#example-ex-4ee604) - [Brandt Modules Over $F_q[t]$](ModBrdt:fldfunrat.md) - [`BrandtModuleDimension(D, N): RngElt, RngElt → RngIntElt`](ModBrdt:fldfunrat.md#function-brandtmoduledimension-rngelt-rngelt) - [`BrandtModuleDimensionOfNewSubspace(D, N): RngElt, RngElt → RngIntElt`](ModBrdt:fldfunrat.md#function-brandtmoduledimensionofnewsubspace-rngelt-rngelt) - [`BrandtModule(M, N): AlgQuatOrd, RngElt → ModBrdt`](ModBrdt:fldfunrat.md#function-brandtmodule-algquatord-rngelt-2) - [`QuaternionOrder(M): ModBrdt → AlgQuatOrd`](ModBrdt:fldfunrat.md#function-quaternionorder-modbrdt) - [`Level(M): ModBrdt → RngElt`](ModBrdt:fldfunrat.md#function-level-modbrdt-2) - [`Discriminant(M): ModBrdt → RngElt`](ModBrdt:fldfunrat.md#function-discriminant-modbrdt-2) - [`Conductor(M): ModBrdt → RngElt`](ModBrdt:fldfunrat.md#function-conductor-modbrdt-2) - [`Ideals(M): ModBrdt → []`](ModBrdt:fldfunrat.md#function-ideals-modbrdt-2) - [`InnerProductMatrix(M): ModBrdt → AlgMatElt`](ModBrdt:fldfunrat.md#function-innerproductmatrix-modbrdt-2) - [`HeckeOperator(M, n): ModBrdtNew, RngElt → Mtrx`](ModBrdt:fldfunrat.md#function-heckeoperator-modbrdtnew-rngelt) - [`HeckeEigenvectors(M): ModBrdt → [ ModBrdt ]`](ModBrdt:fldfunrat.md#function-heckeeigenvectors-modbrdt) - [`HeckeEigenvalue(f, p): ModBrdtElt, RngElt → RngElt`](ModBrdt:fldfunrat.md#function-heckeeigenvalue-modbrdtelt-rngelt)