Fungrim entry: 10f3b2

E_{6}\!\left(\tau\right) = \frac{1}{2} \left(\theta_{3}^{12}\!\left(0, \tau\right) + \theta_{4}^{12}\!\left(0, \tau\right) - 3 \theta_{2}^{8}\!\left(0, \tau\right) \left(\theta_{3}^{4}\!\left(0, \tau\right) + \theta_{4}^{4}\!\left(0, \tau\right)\right)\right)

Assumptions:

\tau \in \mathbb{H}

TeX:

E_{6}\!\left(\tau\right) = \frac{1}{2} \left(\theta_{3}^{12}\!\left(0, \tau\right) + \theta_{4}^{12}\!\left(0, \tau\right) - 3 \theta_{2}^{8}\!\left(0, \tau\right) \left(\theta_{3}^{4}\!\left(0, \tau\right) + \theta_{4}^{4}\!\left(0, \tau\right)\right)\right)

\tau \in \mathbb{H}

Definitions:

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

Source code for this entry:

Entry(ID("10f3b2"),
    Formula(Equal(EisensteinE(6, tau), Mul(Div(1, 2), Sub(Add(Pow(JacobiTheta(3, 0, tau), 12), Pow(JacobiTheta(4, 0, tau), 12)), Mul(Mul(3, Pow(JacobiTheta(2, 0, tau), 8)), Add(Pow(JacobiTheta(3, 0, tau), 4), Pow(JacobiTheta(4, 0, tau), 4))))))),
    Variables(tau),
    Assumptions(Element(tau, HH)))

Fungrim entry: 10f3b2

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