SymbolicModularEtaQuotient QMODΒΆ

qetasymbmod.spad line 256 [edit on github]

SymbolicModularEtaQuotient(QMOD) holds data to compute an eta-quotient expansions of $F_{bar{s}, bar{r}, m, t}(gammatau)$ as several cusps. See eqref{eq:F_sbar-rbar-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 properly bigger than -1. Note that ordersMin(x) gives an upper bound of the order for each cusp. Since x represents a modular function, with only poles at the cusps, 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.

cusps: % -> List Cusp

cusps(x) either returns the value that was given to etaQuotient at creation time or cusps(nn)$QMOD if the cusps where not given where nn is the level given through the cofactor of x.

elt: (%, Cusp) -> SymbolicModularEtaQuotientGamma QMOD

x.cusp returns the data corresponding to the respective cusp.

etaQuotient: (QEtaSpecification, List Cusp) -> %

etaQuotient(rspec,cusps) represents the expansion of $g_{rbar}(gamma tau)$ for all gamma corresponding to the given cusps of QMOD. $rbar$ is given by rspec.

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

etaQuotient(sspec,rspec,m,t) represents the expansion of $F_{rbar,sbar,m,t}(gamma tau)$ for all gamma corresponding to the cusps of QMOD. $rbar$ and $sbar$ are given by rspec and sspec, respectively.

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

etaQuotient(sspec,rspec,m,t,spitzen) represents the expansion of $F_{sbar,rbar,m,t}(gamma tau)$ for all gamma corresponding to the given cusps of QMOD. $rbar$ and $sbar$ are given by rspec and sspec, repectively.

etaQuotient: QEtaSpecification -> %

etaQuotient(rspec) represents the expansion of $g_{rbar}(gamma tau)$ for all gamma corresponding to the cusps of QMOD. $rbar$ is given by rspec.

hash: % -> SingleInteger

from SetCategory

hashUpdate!: (HashState, %) -> HashState

from SetCategory

latex: % -> String

from SetCategory

minimalRootOfUnity: % -> PositiveInteger

minimalRootOfUnity(x) returns lcm [minimalRootOfUnity(x.u) for u in cusps x].

one?: % -> Boolean

one?(x) returns true if the eta-quotient corresponding to x represents 1. This is the case if one?(basefactor(x.c)) and one?(cofactor(x.c)) for one of the cusps of cusps(x).

ordersMin: % -> List Fraction Integer

ordersMin(x) returns an estimate for the order at all cusps (in the canonical variable) without computing the explicit series expansion at any cusp. Note that due to estimation it does not necessarily return an integer, but a rational number. The result is [orderMin(x.c) for c in cusps x].

BasicType

CoercibleTo OutputForm

SetCategory