SymbolicModularEtaQuotient MODG

qetasymbmod.spad line 302 [edit on github]

SymbolicModularEtaQuotient(MODG) 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)}.

=: (%, %) -> Boolean

from BasicType

~=: (%, %) -> Boolean

from BasicType

coerce: % -> OutputForm

from CoercibleTo OutputForm

constant?: % -> Boolean

constant?(x) returns true if 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 x represents 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, x can represent a constant even if constant?(x) is false. If, however, multiplier(x)=1 then x represents 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 x did 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 MODG

x.gamma returns the data corresponding to the respective transformation matrix.

etaQuotient: (QEtaSpecification, List Cusp) -> %

etaQuotient(rspec,gammas) for gammas:=[cuspToMatrix(c)$MODG for c in 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 the group represented by MODG. $r$ and $s$ are given by rspec and sspec, respectively.

etaQuotient: (QEtaSpecification, QEtaSpecification, PositiveInteger, NonNegativeInteger, List Cusp) -> %

etaQuotient(sspec,rspec,m,t,spitzen) computes etaQuotient(sspec,rspec,m,t,gammas) for gammas:=[cuspToMatrix(c)$MODG for c in 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 rspec and sspec, repectively.

etaQuotient: (QEtaSpecificationExpressionMonomial, List Matrix Integer) -> %

etaQuotient(spexmon, gammas) represents the expansion of $F_{s,r,m,t}(gamma tau)$ for all gamma in gammas. $r$ and $s$, $m$, and $t$ are given specificationMonomial(spexmon) if this does not lead to failure.

etaQuotient: QEtaSpecification -> %

etaQuotient(rspec) represents the expansion of $g_r(gamma tau)$ for all gamma corresponding to the cusps of MODG. $r$ is given by rspec.

hash: % -> SingleInteger

from Hashable

hashUpdate!: (HashState, %) -> HashState

from Hashable

latex: % -> String

from SetCategory

minimalRootOfUnity: % -> PositiveInteger

minimalRootOfUnity(x) returns lcm [minimalRootOfUnity(x.u) for u in 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 true if the eta-quotient corresponding to x represents 1. This is the case if one?(basefactor(x.trf)) and one?(cofactor(x.trf)) for one element trf of 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 trf in transformationMatrices(x) the input condition multiplier(basefactor(x.trf))=1 is not met. The result is [order(x.trf) for trf in 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 integers, but rational numbers. The result is [orderMin(x.trf) for trf in transformationMatrices x]. If multiplier(x)=1, then ordersMin(x)=orders(x).

poles: % -> List Matrix Integer

poles(x) returns a list of cusps c (or rather the corresponding transformation matrices that were used at creation time) for which order(x.c)<0. If multiplier(x.c)~=1 for some cusp c, then an error occurs. If this function returns the empty list then the eta-quotient is constant, i.e. equal to 1.

specificationExpressionMonomial: QEtaSpecificationMonomial -> QEtaSpecificationExpressionMonomial

specificationExpressionMonomial(specmon) returns the the corresponding element spexmon of QEtaSpecificationExpressionMonomial such that specmon and spexmon represent the same modular function. The function aborts with an error if specmon does not represent a modular function for MODG.

specificationMonomial: QEtaSpecificationExpressionMonomial -> QEtaSpecificationMonomial

specificationMonomial(x) returns a structure (sspec,rspec,m,t), if specmon=[sspec,rspec,m,t]$QEtaSpecificationMonomial and specificationExpressionMonomial(specmon) is equal to x (modulo renaming of the variables). Otherwise it returns a structure specmon for which failure?(specmon) is true. Note that specificationExpressionMonomial(specmon) implicitly checks whether modular?(sspec,rspec,m,t)$MODG is true, i.e. x must represent a modular function.

transformationMatrices: % -> List Matrix Integer

transformationMatrices(x) either returns the value that was given to etaQuotient at creation time or [cuspToMatrix(level(x),c)$MODG for c in cusps(nn)$MODG] if no transformation matrices or cusps where given.

BasicType

CoercibleTo OutputForm

Hashable

SetCategory