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Fungrim entry: 8ffe07

E2k ⁣(aτ+bcτ+d)=(cτ+d)2kE2k ⁣(τ)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:kZ2andτHand(abcd)SL2(Z)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
EisensteinEEk ⁣(τ)E_{k}\!\left(\tau\right) Normalized Eisenstein series
Powab{a}^{b} Power
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
HHH\mathbb{H} Upper complex half-plane
Matrix2x2(abcd)\begin{pmatrix} a & b \\ c & d \end{pmatrix} Two by two matrix
SL2ZSL2(Z)\operatorname{SL}_2(\mathbb{Z}) Modular group
Source code for this entry:
Entry(ID("8ffe07"),
    Formula(Equal(EisensteinE(Mul(2, k), Div(Add(Mul(a, tau), b), Add(Mul(c, tau), d))), Mul(Pow(Add(Mul(c, tau), d), Mul(2, k)), EisensteinE(Mul(2, k), tau)))),
    Variables(k, tau, a, b, c, d),
    Assumptions(And(Element(k, ZZGreaterEqual(2)), Element(tau, HH), Element(Matrix2x2(a, b, c, d), SL2Z))))

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2019-08-21 11:44:15.926409 UTC