SymbolicEtaQuotient¶
qetaquotsymb.spad line 976 [edit on github]
SymbolicEtaQuotient holds data to compute an eta-quotient expansions of g_r
(tau) at all cusps of $Gamma_0(N
)$ or at the given cusps. See eqref{g_r
(tau)}
- coerce: % -> OutputForm
from CoercibleTo OutputForm
- elt: (%, Cusp) -> SymbolicEtaQuotientGamma
x
.cusp returns the data corresponding to the respective cusp.
- etaQuotient: (PositiveInteger, List Integer) -> %
etaQuotient(mm, r)
represents the expansion ofg_r
(tau) at all cusps a/c ofGamma0
(mm
) in terms ofx
= exp(2 pii
tau/w) where w=width(m
,c
) and gamma=cuspToMatrix(mm
, a/c). It is the same as etaQuotient(mm
,r
, cusps(mm
)$CongruenceSubgroupGamma0
). $r
$ is indexed by the divisors of $mm
$. See eqref{eq:g_r(tau)}.
- etaQuotient: (PositiveInteger, List Integer, List Cusp) -> %
etaQuotient(mm, r, cusps)
represents the expansion ofg_r
(tau) at the given cusps a/c ofGamma0
(mm
) in terms ofx
= exp(2 pii
tau/w) where w=width(m
,c
) and gamma=cuspToMatrix(m
, a/c). $r
$ is indexed by the divisors of $mm
$. See eqref{eq:g_r(tau)}.
- etaQuotient: (QEtaSpecification, List Cusp) -> %
etaQuotient(rspec, cusps)
represents the expansion ofg_r
(tau) at the given cusps a/c ofGamma0
(mm
) in terms ofx
= exp(2 pii
tau/w) where w=width(m
,c
) and gamma=cuspToMatrix(m
, a/c). $r
$ and $mm
$ are given byrspec
. See eqref{eq:g_r(tau)}.
- etaQuotient: QEtaSpecification -> %
etaQuotient(rspec)
represents the expansion ofg_r
(tau) at all cusps a/c ofGamma0
(mm
) in terms ofx
= exp(2 pii
tau/w) where w=width(m
,c
) and gamma=cuspToMatrix(mm
, a/c). It is the same as etaQuotient(rspec
, cusps(mm
)$CongruenceSubgroupGamma0
). $r
$ and $mm
$ are given byrspec
. See eqref{eq:g_r(tau)}.
- exponents: % -> List Integer
If x=etaQuotient(
m
,r
) then exponents(x
) returns the list of exponents, i.e. exponents(x
)=r
.
- hash: % -> SingleInteger
from SetCategory
- hashUpdate!: (HashState, %) -> HashState
from SetCategory
- latex: % -> String
from SetCategory
- level: % -> PositiveInteger
If x=etaQuotient(
mm
,r
) then level(x
) returns the level of the eta-quotient, i.e. level(x
)=mm
.
- minimalRootOfUnity: % -> PositiveInteger
If x=etaQuotient(
mm
,r
), then minimalRootOfUnity(x
) returns the smallest positive integern
such that the expansion of the functiong_r
(tau) at any cusp ofGamma_0
(mm
) (neglecting the (c
tau+d)^*
factor lives inQ
[w
][[x
]] wherew
is ann
-th root of unity.
- one?: % -> Boolean
one?(x)
returnstrue
if the eta-quotient corresponding tox
represents 1.
- qetaGrades: % -> List Integer
qetaGrades(x)
returns the (negated) exponents of a series expansion ofx
at all the cusps in the canonical variables, i.e. in x=q^(1/w) where w=width(levelx
, denom cusp)$GAMMA0
Internally, the matrix of Ligozat is employed.
- specification: % -> QEtaSpecification
If x=etaQuotient(spec,…), then specification(
x
) returns spec.