# Fungrim entry: b1a5e4

$H\!\left(\frac{a \tau + b}{c \tau + d}\right) = {\left(c \tau + d\right)}^{2} H(\tau)\; \text{ where } H(\tau) = G_{2}\!\left(\tau\right) - \frac{\pi}{\operatorname{Im}(\tau)}$
Assumptions:$\tau \in \mathbb{H} \;\mathbin{\operatorname{and}}\; \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \operatorname{SL}_2(\mathbb{Z})$
TeX:
H\!\left(\frac{a \tau + b}{c \tau + d}\right) = {\left(c \tau + d\right)}^{2} H(\tau)\; \text{ where } H(\tau) = G_{2}\!\left(\tau\right) - \frac{\pi}{\operatorname{Im}(\tau)}

\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
Pow${a}^{b}$ Power
EisensteinG$G_{k}\!\left(\tau\right)$ Eisenstein series
Pi$\pi$ The constant pi (3.14...)
Im$\operatorname{Im}(z)$ Imaginary part
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("b1a5e4"),
Assumptions(And(Element(tau, HH), Element(Matrix2x2(a, b, c, d), SL2Z))))