# Fungrim entry: 171724

$E_{6}\!\left(\tau\right) = 1 + 63 \sum_{m=1}^{\infty} \frac{2 \cos^{4}\!\left(\pi m \tau\right) + 11 \cos^{2}\!\left(\pi m \tau\right) + 2}{\sin^{6}\!\left(\pi m \tau\right)}$
Assumptions:$\tau \in \mathbb{H}$
TeX:
E_{6}\!\left(\tau\right) = 1 + 63 \sum_{m=1}^{\infty} \frac{2 \cos^{4}\!\left(\pi m \tau\right) + 11 \cos^{2}\!\left(\pi m \tau\right) + 2}{\sin^{6}\!\left(\pi m \tau\right)}

\tau \in \mathbb{H}
Definitions:
Fungrim symbol Notation Short description
EisensteinE$E_{k}\!\left(\tau\right)$ Normalized Eisenstein series
Sum$\sum_{n} f(n)$ Sum
Pow${a}^{b}$ Power
Cos$\cos(z)$ Cosine
Pi$\pi$ The constant pi (3.14...)
Sin$\sin(z)$ Sine
Infinity$\infty$ Positive infinity
HH$\mathbb{H}$ Upper complex half-plane
Source code for this entry:
Entry(ID("171724"),
Formula(Equal(EisensteinE(6, tau), Add(1, Mul(63, Sum(Div(Add(Add(Mul(2, Pow(Cos(Mul(Mul(Pi, m), tau)), 4)), Mul(11, Pow(Cos(Mul(Mul(Pi, m), tau)), 2))), 2), Pow(Sin(Mul(Mul(Pi, m), tau)), 6)), For(m, 1, Infinity)))))),
Variables(tau),
Assumptions(And(Element(tau, HH))))

## Topics using this entry

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