SymbolicModularEtaQuotient QMOD¶
qetasymbmod.spad line 295 [edit on github]
- QMOD: QEtaModularCategory 
SymbolicModularEtaQuotient(QMOD) holds data to compute an eta-quotient expansions of $F_{s,r,m,t}(gamma tau)$ at several transformation matrices. See eqref{eq:F_s-r-m-t(gamma*tau)}.
- coerce: % -> OutputForm
- from CoercibleTo OutputForm 
- constant?: % -> Boolean
- constant?(x)returns- trueif ordersMin(- x) returns a list of entries that are all properly bigger than- -1. Note that ordersMin(- x) gives an upper bound of the order for each transformation matrix. If- xrepresents a modular function, with only poles at the (stored) transformation matrices of- x, it means that the function must be constant. Since we rely on estimates,- xcan represent a constant even if constant?(- x) is- false. If, however, multiplier(- x)- =1then- xrepresents simply an eta-quotient and ordersMin(- x) gives the exact orders.
- coSpecification: % -> QEtaSpecification
- If x=etaQuotient(sspec,rspec,…) coSpecification( - x) returns sspec. If the definition of- xdid not involve any cofactor, then coSpecification(- x) returns etaQuotientSpecification(level(definingSpecification(- x)),[]).
- definingSpecification: % -> QEtaSpecification
- If x=etaQuotient(sspec,rspec,…) or x=etaQuotient(rspec,spitzen) or x=etaQuotient(rspec), then definingSpecification( - x) returns rspec.
- elt: (%, Matrix Integer) -> SymbolicModularEtaQuotientGamma QMOD
- x.gamma returns the data corresponding to the respective transformation matrix.
- etaQuotient: (QEtaSpecification, List Cusp) -> %
- etaQuotient(rspec,gammas)for nn:=level(- rspec) and gammas:=[cuspToMatrix(- nn,- c)$QMOD for- cin spitzen].
- etaQuotient: (QEtaSpecification, List Matrix Integer) -> %
- etaQuotient(rspec,gammas)represents the expansion of $- g_r(gamma tau)$ for all gamma in- gammas. $- r$ is given by- rspec.
- etaQuotient: (QEtaSpecification, QEtaSpecification, PositiveInteger, NonNegativeInteger) -> %
- etaQuotient(sspec,rspec,m,t)represents the expansion of $- F_{- r,- s,- m,- t}(gamma tau)$ for all gamma corresponding to the cusps of QMOD. $- r$ and $- s$ are given by- rspecand- sspec, respectively.
- etaQuotient: (QEtaSpecification, QEtaSpecification, PositiveInteger, NonNegativeInteger, List Cusp) -> %
- etaQuotient(sspec,rspec,m,t,spitzen)computes etaQuotient(- sspec,- rspec,- m,- t,gammas) for nn:=level(- sspec) and gammas:=[cuspToMatrix(- nn,- c)$QMOD for- cin- spitzen].
- etaQuotient: (QEtaSpecification, QEtaSpecification, PositiveInteger, NonNegativeInteger, List Matrix Integer) -> %
- etaQuotient(sspec,rspec,m,t,gammas)represents the expansion of $- F_{- s,- r,- m,- t}(gamma tau)$ for all gamma in- gammas. $- r$ and $- s$ are given by- rspecand- sspec, repectively.
- etaQuotient: QEtaSpecification -> %
- etaQuotient(rspec)represents the expansion of $- g_r(gamma tau)$ for all gamma corresponding to the cusps of QMOD. $- r$ is given by- rspec.
- hash: % -> SingleInteger
- from Hashable 
- hashUpdate!: (HashState, %) -> HashState
- from Hashable 
- latex: % -> String
- from SetCategory 
- level: % -> PositiveInteger
- level(y)returns level(coSpecification(- y)).
- minimalRootOfUnity: % -> PositiveInteger
- minimalRootOfUnity(x)returns- lcm[minimalRootOfUnity(- x.- u) for- uin transformationMatrices- x].
- multiplier: % -> PositiveInteger
- If x=etaQuotient( - sspec,- rspec,- m,- k,…), then multiplier(- x) returns- m. If x=etaQuotient(- rspec,…), then multiplier(- x)- =1.
- offset: % -> NonNegativeInteger
- If x=etaQuotient( - sspec,- rspec,- m,- k,…), then offset(- x) returns- k. If x=etaQuotient(- rspec,…), then offset(- x)- =0.
- one?: % -> Boolean
- one?(x)returns- trueif the eta-quotient corresponding to- xrepresents 1. This is the case if one?(basefactor(- x.- trf)) and one?(cofactor(- x.- trf)) for one element- trfof transformationMatrices(- x).
- orders: % -> List Integer
- orders(x)returns the order at all transformation matrices (in the canonical variable) without computing the explicit series expansion at any matrix. It returns an error, if for one matrix- trfin transformationMatrices(- x) the input condition multiplier(basefactor(- x.- trf))- =1is not met. The result is [order(- x.- trf) for- trfin transformationMatrices- x].
- ordersMin: % -> List Fraction Integer
- ordersMin(x)returns an estimate for the order at all transformation matrices (in the canonical variable) without computing the explicit series expansion at any matrix. Note that due to estimation it does not necessarily return an integer, but a rational number. The result is [orderMin(- x.- trf) for- trfin transformationMatrices- x].