Fungrim entry: a0dff6

E_{6}^{2}\!\left(\tau\right) = \frac{1}{8} \left({\left(\theta_{2}^{8}\!\left(0, \tau\right) + \theta_{3}^{8}\!\left(0, \tau\right) + \theta_{4}^{8}\!\left(0, \tau\right)\right)}^{3} - 54 {\left(\theta_{2}\!\left(0 , \tau\right) \theta_{3}\!\left(0 , \tau\right) \theta_{4}\!\left(0 , \tau\right)\right)}^{8}\right)

Assumptions:

\tau \in \mathbb{H}

TeX:

E_{6}^{2}\!\left(\tau\right) = \frac{1}{8} \left({\left(\theta_{2}^{8}\!\left(0, \tau\right) + \theta_{3}^{8}\!\left(0, \tau\right) + \theta_{4}^{8}\!\left(0, \tau\right)\right)}^{3} - 54 {\left(\theta_{2}\!\left(0 , \tau\right) \theta_{3}\!\left(0 , \tau\right) \theta_{4}\!\left(0 , \tau\right)\right)}^{8}\right)

\tau \in \mathbb{H}

Definitions:

Fungrim symbol	Notation	Short description
`Pow`	${a}^{b}$	Power
`EisensteinE`	$E_{k}\!\left(\tau\right)$	Normalized Eisenstein series
`JacobiTheta`	$\theta_{j}\!\left(z , \tau\right)$	Jacobi theta function
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("a0dff6"),
    Formula(Equal(Pow(EisensteinE(6, tau), 2), Mul(Div(1, 8), Sub(Pow(Add(Add(Pow(JacobiTheta(2, 0, tau), 8), Pow(JacobiTheta(3, 0, tau), 8)), Pow(JacobiTheta(4, 0, tau), 8)), 3), Mul(54, Pow(Mul(Mul(JacobiTheta(2, 0, tau), JacobiTheta(3, 0, tau)), JacobiTheta(4, 0, tau)), 8)))))),
    Variables(tau),
    Assumptions(Element(tau, HH)))

Topics using this entry

Eisenstein series