# Fungrim entry: 8ffe07

$E_{2 k}\!\left(\frac{a \tau + b}{c \tau + d}\right) = {\left(c \tau + d\right)}^{2 k} E_{2 k}\!\left(\tau\right)$
Assumptions:$k \in \mathbb{Z}_{\ge 2} \;\mathbin{\operatorname{and}}\; \tau \in \mathbb{H} \;\mathbin{\operatorname{and}}\; \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \operatorname{SL}_2(\mathbb{Z})$
TeX:
E_{2 k}\!\left(\frac{a \tau + b}{c \tau + d}\right) = {\left(c \tau + d\right)}^{2 k} E_{2 k}\!\left(\tau\right)

k \in \mathbb{Z}_{\ge 2} \;\mathbin{\operatorname{and}}\; \tau \in \mathbb{H} \;\mathbin{\operatorname{and}}\; \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \operatorname{SL}_2(\mathbb{Z})
Definitions:
Fungrim symbol Notation Short description
EisensteinE$E_{k}\!\left(\tau\right)$ Normalized Eisenstein series
Pow${a}^{b}$ Power
ZZGreaterEqual$\mathbb{Z}_{\ge n}$ Integers greater than or equal to n
HH$\mathbb{H}$ Upper complex half-plane
Matrix2x2$\begin{pmatrix} a & b \\ c & d \end{pmatrix}$ Two by two matrix
SL2Z$\operatorname{SL}_2(\mathbb{Z})$ Modular group
Source code for this entry:
Entry(ID("8ffe07"),
Assumptions(And(Element(k, ZZGreaterEqual(2)), Element(tau, HH), Element(Matrix2x2(a, b, c, d), SL2Z))))