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"),
    Formula(Where(Equal(H(Div(Add(Mul(a, tau), b), Add(Mul(c, tau), d))), Mul(Pow(Add(Mul(c, tau), d), 2), H(tau))), Equal(H(tau), Sub(EisensteinG(2, tau), Div(Pi, Im(tau)))))),
    Variables(tau, a, b, c, d),
    Assumptions(And(Element(tau, HH), Element(Matrix2x2(a, b, c, d), SL2Z))))

Topics using this entry

Eisenstein series