QEtaCofactorConditions¶

undocumented

conditionEvenMultiplier?: (PositiveInteger, QEtaSpecification, PositiveInteger) -> Boolean: conditionEvenMultiplier?(nn, rspec, m) returns true if various divisibility conditions are fulfilled. This corresponds to eqref{eq:even-m} and eqref{eq:conditionEvenMultiplier?(nn,rspec,m)} in qeta.tex fourth condition on page~241 of cite{Radu_RamanujanKolberg_2015} (Definition 35). Same as condition 9 on page 37 of cite{ChenDuZhao_FindingModularFunctionsRamanujan_2019}.

conditionGSigma0?: (PositiveInteger, QEtaSpecification, PositiveInteger) -> Boolean: conditionGSigma0?(nn, rspec, m) returns true if (kappa(m)*nn)*sum(g*e/d for [d,g,e] in properGeneralizedParts rspec) = 0 mod 2. This corresponds to condition 3 on page 37 of Chapter 10 of cite{ChenDuZhao_FindingModularFunctionsRamanujan_2019}. See eqref{eq:conditionGSigma0?(nn,rspec,m)}

conditionMmN?: (PositiveInteger, QEtaSpecification, PositiveInteger) -> Boolean: conditionMmN(nn, rspec, m) returns true if every divisor d of mm=level(rspec) (for which r_d is non-zero) is also a divisor of m*N where r=pureExponents(rspec). This corresponds to condition (4.7) in cite{Radu_PhD_2010} and in equation eqref{eq:delta|M=>delta|mN} in qeta.tex.

conditionNDivisor?: (PositiveInteger, QEtaSpecification, PositiveInteger, NonNegativeInteger) -> Boolean: conditionNDivisor?(nn, rspec, m, t) returns true if the expression (24*m*mm)/gcd(24*mm*kappa(m)*alpha, 24*m*mm) is a divisor of nn where alpha=rhoInfinity(rspec) + t. This corresponds to condition 8 on page 37 of Chapter 10 of cite{ChenDuZhao_FindingModularFunctionsRamanujan_2019} and to eqref{eq:w|N} and eqref{eq:conditionNDivosor?(nn,rspec,m,t)} in qeta.tex and to the third condition on page~241 of cite{Radu_RamanujanKolberg_2015} (Definition 35).

conditionOrbitLength?: (PositiveInteger, QEtaSpecification, PositiveInteger, NonNegativeInteger) -> Boolean: conditionOrbitLength?(nn, rspec, m, t) returns true if the conditions for an orbit of length 1 are satisfied. This corresponds to condition 7 on page 6 of Chapter 2 and to condition 10 on page 37 of Chapter 10 of cite{ChenDuZhao_FindingModularFunctionsRamanujan_2019}.

conditionPrimeDivisors?: (PositiveInteger, PositiveInteger) -> Boolean: conditionExponentSum?(nn, m) returns true if every prime divisor of m is also a divisor of nn. This corresponds to (28) in cite{Radu_RamanujanKolberg_2015} and to eqref{eq:p|m=>p|N} in qeta.tex.

conditionRho0ProperGeneralized?: (PositiveInteger, QEtaSpecification, PositiveInteger) -> Boolean: conditionRho0?(nn, rspec, m) returns true if (kappa(m)*m*nn^2) * sum(e/d for [d,g,e] in properGeneralizedParts x) = 0 mod 12. This corresponds to condition 5 on page 37 of Chapter 10 of cite{ChenDuZhao_FindingModularFunctionsRamanujan_2019}. See eqref{eq:conditionRho0ProperGeneralized?(nn,rspec,m)}.

conditionRho0Pure?: (PositiveInteger, QEtaSpecification, PositiveInteger) -> Boolean: conditionRho0?(nn, rspec, m) returns true if (kappa(m)*m*nn^2/mm)*rho0(rspec) is an integer. This corresponds to eqref{eq:rv24} and eqref{{eq:conditionRho0Pure?(nn,mm,r,m)} in qeta.tex and to the first condition on page~241 of cite{Radu_RamanujanKolberg_2015} (Definition 35).

conditionSumExponentsProperGeneralized?: (PositiveInteger, QEtaSpecification, PositiveInteger) -> Boolean: conditionSumExponentsProperGeneralized?(nn, rspec, m) returns true if the sum of the entries of rspec corresponding to the exponents of properGeneralizedParts(rspec) multiplied by kappa(m)*nn is divisible by 4. This corresponds to condition 4 on page 37 of Chapter 10 of cite{ChenDuZhao_FindingModularFunctionsRamanujan_2019}. See eqref{eq:conditionSumExponentsProperGeneralized?(nn,rspec,m)}.

conditionSumExponentsPure?: (PositiveInteger, QEtaSpecification, PositiveInteger) -> Boolean: conditionSumExponentsPure?(nn, rspec, m) returns true if the sum of the entries of pureExponents(rspec) multiplied by kappa(m)*nn is divisible by 8. This corresponds to eqref{eq:sum-r} in qeta.tex second condition on page~241 of cite{Radu_RamanujanKolberg_2015} (Definition 35).