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Jacobi theta₄ constants at singular nomes
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Numbers
$r$ 
$\theta_4(0,q_r)$
1:
0.9135791381561168214072425934012220897019639163934690334196965312659080093720091139632889833595801389
comment: The square lattice point $\tau_1=i$. Here $\theta_4(0,e^{-\pi})=\theta_2(0,e^{-\pi})$; its Gamma expression is (6).
2:
0.9764760772154493595072986783085003266646376075244065837389351925417586765023178721767366745853714086
3:
0.9913331596853003685864451334686866837806745102347748761730948587873719150506176538536964045288098301
4:
0.9962651145609071357899576385226683356930411621437410809444236442854214076425483840934040291817039305
5:
0.9982209353871232162753675683334362562860953545276380620647762949334070618852299623428509209553834210
6:
0.9990900781129146156272300751905685476927262108710438101116769258913855802322511144200486598886151784
7:
0.9995088326737286280430780775655974690779112863420156641777487943128912545047182783275258073535314380
8:
0.9997233116279591791605045946314341403569695666461571610774763653084774912274284268956157545427665040
9:
0.9998386009648594756245452502537021868110678470493157960903812971903353181303304330429400862671190865
10:
0.9999030622061052905066606663828134093692940745353835438543875093494136499590345474253563280551061614
11:
0.9999403094532222136610367191821282599050852553165889829326473743171593312137859925418887000680544433
12:
0.9999624429333673153612502844692639066369072259805455731701597230261198737944422877265045667375710216
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 $\theta_4(0\mid\tau_r)=\theta_4(0,q_r)=1+2\sum_{n=1}^{\infty}(-1)^nq_r^{n^2}$ in the DLMF convention [1].
Parameters
$r$
—   CM index ($r>0$)
Formulas
(1)
$\theta_4(0,q_r)=\prod_{n=1}^{\infty}(1-q_r^{2n})(1-q_r^{2n-1})^2>0$ [2].
(2)
$\theta_3(0,q_r)^4=\theta_2(0,q_r)^4+\theta_4(0,q_r)^4$, where $\theta_2(0,q)=2\sum_{n=0}^{\infty}q^{(n+1/2)^2}$ and $\theta_3(0,q_r)$ is the companion constant $\theta_3(0,q)=1+2\sum_{n=1}^{\infty}q^{n^2}$ [1] [4].
(3)
For $m_r=(\theta_2(0,q_r)/\theta_3(0,q_r))^4$ in the singular-modulus table, $\theta_4(0,q_r)=(1-m_r)^{1/4}\sqrt{2K(m_r)/\pi}$ and $K(1-m_r)=\sqrt r\,K(m_r)$. Here $K(m_r)$ is in the singular elliptic-integral table, and $K(m)=\int_0^{\pi/2}(1-m\sin^2 t)^{-1/2}\,dt$ uses the parameter $m$ [5] [6].
(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 $Q_r=q_r^2=e^{2\pi i\tau_r}$ [7].
(5)
$\theta_2(0,q_r)\theta_3(0,q_r)\theta_4(0,q_r)=2\eta(\tau_r)^3$ [8] [6].
(6)
$\theta_4(0,e^{-\pi})=\theta_2(0,e^{-\pi})=\Gamma(1/4)/(2\pi)^{3/4}$, where $\Gamma(1/4)$ is the Gamma value at one quarter. This follows from $\lambda(i)=1/2$, $\eta(i)=\Gamma(1/4)/(2\pi^{3/4})$ [7], and (5).
(7)
$\theta_4(0,q_r)=(1-m_r)^{1/4}\mathfrak f(\tau_r)^2\eta(\tau_r)=\sqrt2\,(1-m_r)^{1/4}G_r^2\eta(\tau_r)$, 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]; (2) gives the remaining factor.
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.
(9)
Write $m_r=k_r^2=\lambda(\tau_r)$ for the elliptic parameter in the singular-modulus table. The complete elliptic integral $K(m)$ 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-four table at rational nomes stores the same function with the nome itself as its parameter. Here $r$ indexes the rectangular CM points $\tau_r=i\sqrt r$.
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)[3].real()
Links
Similar tables
Jacobi theta-three constants at singular nomes —   The companion constants at the same CM indices satisfy (2) and (5).
$\theta_4(0,q)$ at rational nomes —   This is the same theta constant indexed by rational nomes; the present table is indexed by the positive integers specifying rectangular CM points.
Singular values $k_r$ of the elliptic modulus —   The elliptic parameters $m_r=k_r^2$ determine the ratio $\theta_4(0,q_r)/\theta_3(0,q_r)=(1-m_r)^{1/4}$.
$K(m_r)$ at singular values —   The complete elliptic integrals at the same CM indices give $\theta_4(0,q_r)$ by (3).
Complete elliptic integral $K(m)$ —   This is the same elliptic integral at rational parameters; its $m=1/2$ entry corresponds to $r=1$ in (3).
Ramanujan and Weber class invariants —   The class invariants at the same lattice parameters determine $\theta_4(0,q_r)/\eta(\tau_r)$ 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 $K(m_r)$.
Data properties
Entries are of type: real number
digits: 100 (Unknown key)
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 426 bits of accuracy, more than 128 decimal digits, and its entire ball lies inside the interval represented by the stored 100-digit decimal.

more

The attached checks compare every row with elementary theta series and the theta-four product at 512 bits, with explicit bounds for the omitted terms and factors. They check the quartic and eta identities, the elliptic relations against T295 and T296, the Ramanujan and Weber relation against T297, and the companion theta-three values against T504. The square-point Gamma identity is an independent special-value check. Every comparison requires finite enclosures with radii less than $10^{-90}$ and overlap; known-value and failure controls precede the comparisons. The checks are repeated on the stored values.



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