Fungrim entry: 0a5ef4

E_{6}\!\left(\tau\right) = \frac{\eta^{24}\!\left(\tau\right)}{\eta^{12}\!\left(2 \tau\right)} - 480 \eta^{12}\!\left(2 \tau\right) - 16896 \frac{\eta^{12}\!\left(2 \tau\right) \eta^{8}\!\left(4 \tau\right)}{\eta^{8}\!\left(\tau\right)} + 8192 \frac{\eta^{24}\!\left(4 \tau\right)}{\eta^{12}\!\left(2 \tau\right)}

Assumptions:

\tau \in \mathbb{H}

References:

K. Ono (2004), Web of Modularity: Arithmetic of the Coefficients of Modular Forms and q-series, American Mathematical Society. Theorem 1.67.

TeX:

E_{6}\!\left(\tau\right) = \frac{\eta^{24}\!\left(\tau\right)}{\eta^{12}\!\left(2 \tau\right)} - 480 \eta^{12}\!\left(2 \tau\right) - 16896 \frac{\eta^{12}\!\left(2 \tau\right) \eta^{8}\!\left(4 \tau\right)}{\eta^{8}\!\left(\tau\right)} + 8192 \frac{\eta^{24}\!\left(4 \tau\right)}{\eta^{12}\!\left(2 \tau\right)}

\tau \in \mathbb{H}

Definitions:

Fungrim symbol	Notation	Short description
`EisensteinE`	$E_{k}\!\left(\tau\right)$	Normalized Eisenstein series
`Pow`	${a}^{b}$	Power
`DedekindEta`	$\eta(\tau)$	Dedekind eta function
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("0a5ef4"),
    Formula(Equal(EisensteinE(6, tau), Add(Sub(Sub(Div(Pow(DedekindEta(tau), 24), Pow(DedekindEta(Mul(2, tau)), 12)), Mul(480, Pow(DedekindEta(Mul(2, tau)), 12))), Mul(16896, Div(Mul(Pow(DedekindEta(Mul(2, tau)), 12), Pow(DedekindEta(Mul(4, tau)), 8)), Pow(DedekindEta(tau), 8)))), Mul(8192, Div(Pow(DedekindEta(Mul(4, tau)), 24), Pow(DedekindEta(Mul(2, tau)), 12)))))),
    Variables(tau),
    Assumptions(Element(tau, HH)),
    References("K. Ono (2004), Web of Modularity: Arithmetic of the Coefficients of Modular Forms and q-series, American Mathematical Society. Theorem 1.67."))

Topics using this entry

Eisenstein series