NumberDB Advanced search Tags Tables Help Login

Jacobi theta₂ constants at singular nomes

edit · history · discussion · files · short url · special functions special values
Numbers
$r$ 
$\theta_2(0,q_r)$
1:
0.9135791381561168214072425934012220897019639163934690334196965312659080093720091139632889833595801389
comment: The square lattice point $\tau_1=i$. Here $\theta_2(0,e^{-\pi})=\theta_4(0,e^{-\pi})$; its Gamma expression is (6).
equals: $\theta_4(0,e^{-\pi})$
2:
0.6587341636868087114484261991426182184349575170028492178187130533378531872650249578916285849645708538
3:
0.5131518035364283464167427809760052571776538945486147049080744270687965521522857125320935924865257047
4:
0.4157606025960270323145071362847439246488730735296086805037924302108541045187552587408754717154039373
5:
0.3453987573693434832583000411340316724513970888092195679018678012366022698024484844024188724196373632
6:
0.2920946580933849999565707721117488853240395779706612687299894416667478544001004140328076800796900503
7:
0.2503686168586032653155820655529367929714372945994953613109532944445561939516831998547900141318224189
8:
0.2169053340203222751552657389061781609332423912759947178996210409267366889068411086814318696918296287
9:
0.1895604509188054786269057033517587720577184337508730829420099982478710160052243883307350800352772271
10:
0.1668767298832390126873437190864130890084448752445431641467852350868675295540134571348652593501753058
11:
0.1478253399538909383837589438432730622747909591506827947590785289273691681821754813158693272114665062
12:
0.1316574420690201027167240791521530344736248490240303436009452012243984162774678180526971443251004580
Definition
For a positive integer $r$, set $\tau_r=i\sqrt r$ and $q_r=e^{\pi i\tau_r}=e^{-\pi\sqrt r}$, taking the positive square root. The entry is the full Jacobi theta constant $\theta_2(0\mid\tau_r)=\theta_2(0,q_r)=2\sum_{n=0}^{\infty}q_r^{(n+1/2)^2}$ in the DLMF convention [1].
Parameters
$r$
—   CM index ($r>0$)
Formulas
(1)
$\theta_2(0,q_r)=2q_r^{1/4}\prod_{n=1}^{\infty}(1-q_r^{2n})(1+q_r^{2n})^2>0$ [2].
(2)
$\theta_3(0,q_r)^4=\theta_2(0,q_r)^4+\theta_4(0,q_r)^4$, where $\theta_3(0,q_r)$ and $\theta_4(0,q_r)$ are the companion theta constants in the same DLMF convention [1] [4].
(3)
For $m_r=(\theta_2(0,q_r)/\theta_3(0,q_r))^4$ in the singular-modulus table, $\theta_2(0,q_r)=m_r^{1/4}\sqrt{2K(m_r)/\pi}$, $K(m_r)=\pi\theta_3(0,q_r)^2/2$ and $K(1-m_r)=\sqrt r\,K(m_r)$. Here $K(m_r)$ is the complete elliptic integral with the parameter convention [5].
(4)
Dedekind's eta function satisfies $\eta(\tau_r)=q_r^{1/12}\prod_{n=1}^{\infty}(1-q_r^{2n})=Q_r^{1/24}\prod_{n=1}^{\infty}(1-Q_r^n)>0$, where $Q_r=q_r^2$ [8].
(5)
$\theta_2(0,q_r)\theta_3(0,q_r)\theta_4(0,q_r)=2\eta(\tau_r)^3$ [6] [7].
(6)
$\theta_2(0,e^{-\pi})=\theta_4(0,e^{-\pi})=\Gamma(1/4)/(2\pi)^{3/4}$, where $\Gamma(1/4)$ is the Gamma value at one quarter. This follows from $\lambda(i)=1/2$, $\eta(i)=\Gamma(1/4)/(2\pi^{3/4})$ [8], and (5).
(7)
$\theta_2(0,q_r)=m_r^{1/4}\mathfrak f(\tau_r)^2\eta(\tau_r)=\sqrt2\,m_r^{1/4}G_r^2\eta(\tau_r)$, where $G_r$ and $\mathfrak f(\tau_r)=2^{1/4}G_r$ are Ramanujan and Weber class invariants. Their defining products give $\mathfrak f(\tau_r)^2\eta(\tau_r)=\theta_3(0,q_r)$ [3]; (3) gives the factor $m_r^{1/4}$.
Comments
(8)
The lattice $\mathbb Z+\mathbb Z i\sqrt r$ has complex multiplication by the order $\mathbb Z[\sqrt{-r}]$, of discriminant $-4r$. Retaining square factors of $r$ retains this lattice and order. The word singular describes the nome at a quadratic imaginary lattice parameter; it does not assert that the theta constant is algebraic. These indices describe rectangular CM points, rather than all CM points or all ideal classes.
(9)
Write $m_r=k_r^2=\lambda(\tau_r)$ for the elliptic parameter in the singular-modulus table. The complete elliptic integral $K(m)=\int_0^{\pi/2}(1-m\sin^2 t)^{-1/2}\,dt$ takes the parameter $m$. DLMF writes this same integral as $K(k)$ with modulus $k=\sqrt m$ [5]. All roots in the formulas are positive real roots.
(10)
The theta convention uses $q_r=e^{\pi i\tau_r}$; the Dedekind eta product is often written using $Q_r=q_r^2=e^{2\pi i\tau_r}$ [8].
Programs
(P1)
Sage
import numberdb.sage as numberdb
from sage.rings.complex_arb import ComplexBallField

r = 1
C = ComplexBallField(numberdb.bits(100, losing=96))
tau = C.gen(0) * C(r).sqrt()
C(0).jacobi_theta(tau)[1].real()
Links
Similar tables
Jacobi theta-three constants at singular nomes —   The companion constants at the same CM indices satisfy (2) and (5).
Jacobi theta-four constants at singular nomes —   The companion constants satisfy (2) and (5); their $r=1$ value equals the theta-two value.
Dedekind eta values at rectangular CM points —   The eta values at the same CM indices enter (5) and (7).
Singular values of the elliptic modulus —   The elliptic parameters satisfy $m_r=k_r^2=(\theta_2(0,q_r)/\theta_3(0,q_r))^4$.
$K(m_r)$ at singular values —   The complete elliptic integrals at the same CM indices give $\theta_2(0,q_r)$ by (3).
Complete elliptic integral $K(m)$ —   The same elliptic integral at rational parameters includes $m=1/2$, which corresponds to $r=1$ in (3).
Ramanujan and Weber class invariants —   The class invariants at the same lattice parameters determine $\theta_2(0,q_r)/\eta(\tau_r)$ together with $m_r$, by (7).
Elliptic alpha function $\alpha(r)$ —   The elliptic alpha function uses the same CM index $r$, elliptic parameter $m_r$ and integral $K(m_r)$.
$\theta_4(0,q)$ at rational nomes —   That table evaluates the fourth theta constant with rational nome $q$ as its parameter. Here the integer CM index $r$ determines the nome $q_r$; (2) relates the theta constants at a common nome.
Data properties
Entries are of type: real number
How well the digits are known: proven
Table is complete: no (it holds every integer $1\leq r\leq12$, the first small rectangular CM cases, including the square lattice and the square-index changes of scale)
How they were obtained:

The generator evaluates the positive square root in $\tau_r$ and the full theta constant using Arb complex balls at 429 bits for 100 requested significant digits. Every value retains at least 423 bits of accuracy, more than 127 decimal digits, and its entire ball lies inside the interval represented by the stored 100-digit decimal.

more

The attached independent checks use real balls at 512 bits, elementary theta series through $n=12$ with geometric tail bounds, and the theta-two and eta products through 80 factors with explicit omitted-product bounds. They check the quartic and eta identities, the elliptic relations against T295 and T296, the Ramanujan and Weber relations against T297, and the companion values against T504, T505 and T506. The square-point Gamma identity is a special-value check outside the theta family. Every comparison requires finite enclosures with radii less than $10^{-90}$ and overlap. Known-value controls and controls that must fail precede the comparisons. The checks are repeated on the stored values.



API Python package Skill for AI Privacy Legal notice Source
Data licensed CC BY-SA 4.0. No tracking, no analytics, essential cookies only.