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Fungrim entry: 0a5ef4

E6 ⁣(τ)=η24 ⁣(τ)η12 ⁣(2τ)480η12 ⁣(2τ)16896η12 ⁣(2τ)η8 ⁣(4τ)η8 ⁣(τ)+8192η24 ⁣(4τ)η12 ⁣(2τ)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:τH\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
EisensteinEEk ⁣(τ)E_{k}\!\left(\tau\right) Normalized Eisenstein series
Powab{a}^{b} Power
DedekindEtaη ⁣(τ)\eta\!\left(\tau\right) Dedekind eta function
HHH\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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2019-09-22 15:43:45.410764 UTC