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Fungrim entry: 21839d

λ ⁣(aτ+bcτ+d)=λ ⁣(τ)\lambda\!\left(\frac{a \tau + b}{c \tau + d}\right) = \lambda\!\left(\tau\right)
Assumptions:τHand(abcd)SL2(Z)and(abcd)(1001)(mod2)\tau \in \mathbb{H} \,\mathbin{\operatorname{and}}\, \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \operatorname{SL}_2(\mathbb{Z}) \,\mathbin{\operatorname{and}}\, \begin{pmatrix} a & b \\ c & d \end{pmatrix} \equiv \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \pmod {2}
TeX:
\lambda\!\left(\frac{a \tau + b}{c \tau + d}\right) = \lambda\!\left(\tau\right)

\tau \in \mathbb{H} \,\mathbin{\operatorname{and}}\, \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \operatorname{SL}_2(\mathbb{Z}) \,\mathbin{\operatorname{and}}\, \begin{pmatrix} a & b \\ c & d \end{pmatrix} \equiv \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \pmod {2}
Definitions:
Fungrim symbol Notation Short description
ModularLambdaλ ⁣(τ)\lambda\!\left(\tau\right) Modular lambda function
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("21839d"),
    Formula(Equal(ModularLambda(Div(Add(Mul(a, tau), b), Add(Mul(c, tau), d))), ModularLambda(tau))),
    Variables(tau, a, b, c, d),
    Assumptions(And(Element(tau, HH), Element(Matrix2x2(a, b, c, d), SL2Z), CongruentMod(Matrix2x2(a, b, c, d), Matrix2x2(1, 0, 0, 1), 2))))

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2019-09-15 14:14:26.267625 UTC