# 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."))

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Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2021-03-15 19:12:00.328586 UTC