QEtaModularGamma0 nn

qetamod.spad line 703 [edit on github]

QEtaModularGamma0 provides functions to check modularity for Gamma0. TODO: matrixEtaOrderElement(cusp,idx)=a_nn(c,delta)/24 where a_nn(di,dj) given by Notation 3.2.6 in cite{Ligozat_CourbesModulaires_1975} where di and dj run over all positive divisors of nn. The Ligozat matrix is A_N as described after Lemma 5.2 in cite{HemmeckeRadu_EtaRelations_2019}. The parameter idx corresponds to the index of a pure eta-quotient, i.e. it is of the form idx=[delta] with delta a divisor of nn. We assume that cusp=(a:c). matrixModular(idxs) returns part of the matrix matrixModular() corresponding to the divisors that are given through idxs (i.e. the order of the columns corresponds to what is given through idxs). Irrelevant rows in the rows specifying the square condition are removed. nn represents the level of $Gamma_0(nn)$ and idxs has the form [[d1],…,[dn]] where the di are (different) divisors of nn. See description for matrixModular() for more information. modular?(nn,sspec,rspec,m,t) returns true iff all the conditions of Theorem~ref{thm:RaduConditions} are fulfilled. Compare with modularGamma0? from QEtaAuxiliaryPackage.

cuspMatrices: () -> List Matrix Integer

from QEtaModularGammaCategory

cusps: () -> List Cusp

from QEtaModularGammaCategory

cuspToMatrix: Cusp -> Matrix Integer

from QEtaModularGammaCategory

etaCofactorSpace: (List List Integer, QEtaSpecification, PositiveInteger, NonNegativeInteger) -> Record(indices: List List Integer, particular: Union(Vector Integer, failed), basis: List Vector Integer)

from QEtaModularGammaCategory

etaCofactorSpaceSystem: (List List Integer, QEtaSpecification, PositiveInteger, NonNegativeInteger) -> Record(qmat: Matrix Fraction Integer, zrels: Vector Integer, qrhs: Vector Fraction Integer, qsgn: List Integer)

from QEtaModularGammaCategory

etaQuotientMonoidInfinitySystem: List List Integer -> Record(qmat: Matrix Fraction Integer, zrels: Vector Integer, qrhs: Vector Fraction Integer, qsgn: List Integer)

from QEtaModularGammaCategory

etaQuotientMonoidNoPolesSystem: (List List Integer, List Cusp) -> Record(qmat: Matrix Fraction Integer, zrels: Vector Integer, qrhs: Vector Fraction Integer, qsgn: List Integer)

from QEtaModularGammaCategory

etaQuotientMonoidSystem: (List List Integer, List Cusp) -> Record(qmat: Matrix Fraction Integer, zrels: Vector Integer, qrhs: Vector Fraction Integer, qsgn: List Integer)

from QEtaModularGammaCategory

etaQuotientMonoidSystem: (List List Integer, List Cusp, List Cusp) -> Record(qmat: Matrix Fraction Integer, zrels: Vector Integer, qrhs: Vector Fraction Integer, qsgn: List Integer)

from QEtaModularGammaCategory

etaQuotientMonoidSystem: (List List Integer, List Cusp, List Integer, List Integer, List Cusp) -> Record(qmat: Matrix Fraction Integer, zrels: Vector Integer, qrhs: Vector Fraction Integer, qsgn: List Integer)

from QEtaModularGammaCategory

genus: () -> NonNegativeInteger

from QEtaModularGammaCategory

groupLevel: () -> PositiveInteger

from QEtaModularGammaCategory

matrixEtaOrder: (List Cusp, List List Integer) -> Matrix Fraction Integer

from QEtaModularGammaCategory

matrixEtaOrderElement: (Cusp, List Integer) -> Fraction Integer

from QEtaModularGammaCategory

matrixEtaOrderFull: (List Cusp, List List Integer) -> Matrix Fraction Integer

from QEtaModularGammaCategory

matrixEtaOrderRow: (Cusp, List List Integer) -> List Fraction Integer

from QEtaModularGammaCategory

matrixLigozat: () -> Matrix Integer

matrixLigozat() returns the (n,n) matrix with entries a_nn(di,dj) given by Notation 3.2.6 in cite{Ligozat_CourbesModulaires_1975} where di and dj run over all n positive divisors of nn. This matrix is A_N as described after Lemma 5.2 in cite{HemmeckeRadu_EtaRelations_2019}.

matrixLigozatElement: (Integer, Integer) -> Integer

matrixLigozatElement(c, delta) computes an entry of the matrix of Ligozat corresponding to the index (c, delta) counted in divisors of nn.

matrixModular: List List Integer -> Matrix Fraction Integer

from QEtaModularGammaCategory

modular?: (QEtaSpecification, QEtaSpecification, PositiveInteger, NonNegativeInteger) -> Boolean

from QEtaModularGammaCategory

modular?: QEtaSpecification -> Boolean

from QEtaModularGammaCategory

modularOrbit: (QEtaSpecification, PositiveInteger, NonNegativeInteger) -> List NonNegativeInteger

from QEtaModularGammaCategory

modularOrbit: QGeneratingFunctionVariable -> List NonNegativeInteger

from QEtaModularGammaCategory

numberOfGapsForSamba: () -> Integer

from QEtaModularGammaCategory

rightCosetRepresentatives: PositiveInteger -> List Matrix Integer

from QEtaModularGammaCategory

weaklyModular?: QEtaSpecification -> Boolean

from QEtaModularGammaCategory

width: (PositiveInteger, Matrix Integer) -> PositiveInteger

from QEtaModularGammaCategory

width: Cusp -> PositiveInteger

from QEtaModularGammaCategory

width: Matrix Integer -> PositiveInteger

from QEtaModularGammaCategory

QEtaModularGammaCategory