Values of the Jacobi theta constant $\theta_4(0,q)$

edit · history · discussion · files · long url · special functions special values

Numbers

$q$

$\theta_4(0,q)$

1/12:

0.8334297835630038958325661370930199456905125963649336332355687430270559458866502069232443435000068981

1/11:

0.8183184200246960027374176396076481030499040349169078121268321935630692623126979489788826814159479965

1/10:

0.8001999980000001999999998000000000019999999999998000000000000001999999999999999980000000000000000002

1/9:

0.7780826041959810188312347574856528713974090775784935666682489102796533170479134745142500374953928906

1/8:

0.7504882663488459115796484114426614804473060390380495869085701107847114896698560294730153953926265640

1/7:

0.7151186509796368025117958539333924150356969387709416242595227903833795241921855402988117765506442405

1/6:

0.6682096780858585622606834636524879509757692550959403874706155047104141885045160264334271940147744445

2/11:

0.6385488451476958844151005054906132345781759903604787569526510765900006720667908207788279541988748949

1/5:

0.6031989760131071932891137374389533594100093194269488147370746199215417368312953416453486081423271089

2/9:

0.5604302177921309239540981455376128518838922919777067160231460267642301569350519871852687428932087075

1/4:

0.5078048710711282609513233740293515851229745932483153161157165322098564101857772116646898770311327244

3/11:

0.4655935793691523371278127422427349177070600130332747742824285776248078951804048889377125176569300753

2/7:

0.4418738369324509632750150528175875562721796229052388244389537648535537249459508829759475962750697680

3/10:

0.4161606426091747425783006227341330826132572536329205485697821050247987890447966784484593879125090067

1/3:

0.3579231272899599030259112341037224646714049513082803954163690883300487501132413258131403699496238938

4/11:

0.3074753992710443427881226507896631089756825494611182525558282331125383273479507533977852957139168662

3/8:

0.2892577875147163845245444172486432795724538794980206251688251755429727014205900129323330625255598783

2/5:

0.2506765707682886633010584773354776141000802727902141772792608923633519331067701894545775905983581444

5/12:

0.2261928950560674802702310075838481265640795551387850157194514179386226879545463537237741505934042765

3/7:

0.2093560927360337799729536385095343273347350089347093084077452807773258851492096970081366132115008346

4/9:

0.1877993492669674752083623867686985171619456975665243673780121034107725599746426256338707632181921746

5/11:

0.1746357775217822448006250791658700772653314971159208610301930057144421068986958336468514555844368892

1/2:

0.1211242080025805024608492931818675058098582468209605972339011083968975317975043492585129661653781781

6/11:

0.07770240998590889898699010512568912690063571362903806680982269622635798255076873184898904222031685166

5/9:

0.06948977474661941865368232505769363252759011697457684955421173638717609709951734128669997612100840052

4/7:

0.05765179425721636269177679870491231698166597580918999027170873351324825770931432646636815774456167174

7/12:

0.04962636240101650305773390658595701393578019555072604127765111268440240555455603419716848564897815814

3/5:

0.03960316452580476770515475621748900815665704734053002686473260341820335227395832452617004477543280985

5/8:

0.02714046523178527333924621138084208976914352427967847666638016368028317567792920883614573046732927710

7/11:

0.02244995811641888650222743007961771254964557858009638492877427176849411663975363504145818550587360778

2/3:

0.01267038972657299063966092272337368274742829896875516769985713306716097371230379968051922221066701449

7/10:

0.005876410710348859157893764190435159857945745799669721143403329764842247911520809497746706288845012322

5/7:

0.003993779635221903652257572075237356304183846619784937671526105457622235460980204240115723063691463064

8/11:

0.002711057131004092994502138041419146941653936412284706004885753618336631924873192440962719988887836904

3/4:

0.001245309430198100883360550765546521830048696502199748600896527283470485875320670908993661570341287219

7/9:

0.0003851233358290454855786185661850030871683341124082171723415470091441801087115642327793290723280156194

4/5:

0.0001183364390046378921420838455300162320090761397734077763426269088121484615102306895231350330540576303

9/11:

0.00003617237821891970255962414018878658743830577341681677431332066056192326583509454210616209670733291519

5/6:

0.00001100969141047105484491613307015586384386171416672622742164777597391967527628195206042980071903936416

6/7:

0.000001009556680423339826607574576840782044420660344913638397559238397626748496516076764784804938823691512

7/8:

9.159962553050060627076014464728130448771344341406858125626816118607830293120271134852791691534714328e-8

8/9:

8.244420679456433483617401201300823649609182116606161804450523793094741612573331810890560108752319450e-9

9/10:

7.373526938473068047969135475469319468124353000395636092859067650315387147885007239936682990623263839e-10

-1/2:

2.128936827211877158669458548544951324612516539940878092889011192056960266359156813318261280324938297

-1/3:

1.691459681681715341348423255090626896558465260574686790493484256757898066651787223619023063555211517

Definition

For a real nome $-1<q<1$, this table stores the Jacobi theta constant $\theta_4(0,q)$, with $q=e^{i\pi\tau}$ and $\theta_4(z,q)=1+2\sum_{n=1}^{\infty}(-1)^nq^{n^2}\cos(2nz)$ [1] [2].

Parameters

$q$

— real nome ($-1<q<1$)

Formulas

(1)

$\theta_4(0,q)=1+2\sum_{n=1}^{\infty}(-1)^nq^{n^2}$.

(2)

$\theta_3(0,q)^4=\theta_2(0,q)^4+\theta_4(0,q)^4$, where $\theta_2(0,q)=2\sum_{n=0}^{\infty}q^{(n+1/2)^2}$ and $\theta_3(0,q)=\sum_{n\in\mathbb Z}q^{n^2}$ [2] [3].

(3)

$\theta_4(0,-q)=\theta_3(0,q)$ for real $|q|<1$, because $n^2$ and $n$ have the same parity in the defining series. Also $\theta_4(0,0)=1$ [6].

Comments

(4)

The parameter is the real nome $q$, not the lattice parameter $\tau$. In this rectangular case $0<q<1$, the relation $q=e^{i\pi\tau}$ determines $\tau=\log(q)/(i\pi)$ uniquely [1]. Some modular-form sources write $q$ for $e^{2\pi i\tau}$, which is the square of this nome [5].

(5)

These are theta constants: $z$ is fixed at $0$ and is not a parameter. The value $\theta_1(0,q)$ is $0$ for every $q$, because $\theta_1(z,q)$ is odd in $z$ [2].

(6)

Rational nomes here and the singular nomes in Jacobi_theta₄_constants_at_singular_nomes use different arguments. The singular-nome table indexes $q=e^{-\pi\sqrt r}$ by the positive integer $r$. It does not replace this rational-nome parameterization [1].

(7)

The earlier positive-nome rows are rectangular cases. Negative nomes can instead be represented by $\tau=1-i\log|q|/\pi$; the row $q=0$ is the convergent-series limit.

Programs

(P1)

Sage

import numberdb.sage as numberdb
from sage.rings.complex_arb import ComplexBallField
from sage.rings.rational_field import QQ

CBF = ComplexBallField(numberdb.bits(100, losing=80))
q = QQ(1) / QQ(10)
tau = CBF(0, -1) * CBF(q).log() / CBF.pi()
value = CBF(0).jacobi_theta(tau)[3]
value.real()

Links

[1]

DLMF 20.1: Special notation for theta functions

[2]

DLMF 20.2: Definitions and periodic properties of theta functions

[3]

DLMF 20.7: Identities for theta functions

[4]

DLMF 20.9: Relations of theta functions to other functions

[5]

Wikipedia: Modular form

[6]

DLMF 20.2: defining theta series

Similar tables

Values of the elliptic nome $q(m)$ — it stores $q(m)$ at rational elliptic parameters $m$, not at the rational nomes used here. The identity $\theta_4(0,q(m))^2/\theta_3(0,q(m))^2=\sqrt{1-m}$ relates the functions, not their selected input lists [4]

Complete elliptic integral of the first kind $K(m)$ — Jacobi's inversion formula gives $K(m)=\frac{\pi}{2}\theta_3(0,q)^2$ when $q$ is the nome of $m$ [4]

Singular values $k_r$ of the elliptic modulus — it stores $k_r=\theta_2(0,q)^2/\theta_3(0,q)^2$ at $q=e^{-\pi\sqrt r}$; at the same nome $\theta_4(0,q)^2/\theta_3(0,q)^2$ is the complementary modulus [4]

Complete elliptic integral of the first kind $K(m_r)$ at the singular values — it stores $K(m_r)=\frac{\pi}{2}\theta_3(0,e^{-\pi\sqrt r})^2$, the singular-nome case of Jacobi's inversion formula [4]

Values of the elliptic alpha function $\alpha(r)$ — its defining formula uses $\theta_4(0,q)$ and its $q$-derivative at the singular nome $q=e^{-\pi\sqrt r}$

Data properties

Entries are of type: real number

How well the digits are known: proven

Table is complete: no (it retains the $43$ reduced positive rational nomes of denominator at most $12$ with $q\leq9/10$ as a bounded exploratory reference, distinct from the singular-nome family. The exact degeneration $q=0$ and the short negative-nome slice $q=-1/2,-1/3$ extend coverage through the parity relation $\theta_4(0,-q)=\theta_3(0,q)$, rather than extending a rational grid)

How they were obtained:

Each value was computed as a Sage complex ball with arb at numberdb.bits(digits, losing=80) bits and compared against the defining $q$-series with an explicit tail bound and against (2) on every row. The nome was converted to $\tau=\log(q)/(i\pi)$; the imaginary part was checked to contain zero and the real part was returned.

The coverage extension retains all previous numeric strings and entry metadata. The complete numerical selection is recomputed by the attached standalone Python generator using Arb at 640 bits, including rational and period-argument uncertainty. Exact anchors remain exact. Every number is independently compared with mpmath at 160 and 210 decimal digits; new integral values also pass defining-integral quadrature and reduction identities where applicable. New division-point values pass inverse-integral checks, and the half-quarter-period values also pass their radical formula. The coverage-checks.json attachment identifies the checks actually applicable to each row. Existing comments and referral branches retain their earlier provenance.