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Fungrim entry: fc3ef5

(εj ⁣(a,b,c,d))4=(1)n   where n={a(b+d)+cd,j=1a(b+d),j=2ad,j=3d(a+c),j=4{\left(\varepsilon_{j}\!\left(a, b, c, d\right)\right)}^{4} = {\left(-1\right)}^{n}\; \text{ where } n = \begin{cases} a \left(b + d\right) + c d, & j = 1\\a \left(b + d\right), & j = 2\\a d, & j = 3\\d \left(a + c\right), & j = 4\\ \end{cases}
Assumptions:j{1,2,3,4}and(abcd)SL2(Z)j \in \left\{1, 2, 3, 4\right\} \,\mathbin{\operatorname{and}}\, \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \operatorname{SL}_2(\mathbb{Z})
TeX:
{\left(\varepsilon_{j}\!\left(a, b, c, d\right)\right)}^{4} = {\left(-1\right)}^{n}\; \text{ where } n = \begin{cases} a \left(b + d\right) + c d, & j = 1\\a \left(b + d\right), & j = 2\\a d, & j = 3\\d \left(a + c\right), & j = 4\\ \end{cases}

j \in \left\{1, 2, 3, 4\right\} \,\mathbin{\operatorname{and}}\, \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \operatorname{SL}_2(\mathbb{Z})
Definitions:
Fungrim symbol Notation Short description
Powab{a}^{b} Power
JacobiThetaEpsilonεj ⁣(a,b,c,d)\varepsilon_{j}\!\left(a, b, c, d\right) Root of unity in modular transformation of Jacobi theta functions
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("fc3ef5"),
    Formula(Equal(Pow(JacobiThetaEpsilon(j, a, b, c, d), 4), Where(Pow(-1, n), Equal(n, Cases(Tuple(Add(Mul(a, Add(b, d)), Mul(c, d)), Equal(j, 1)), Tuple(Mul(a, Add(b, d)), Equal(j, 2)), Tuple(Mul(a, d), Equal(j, 3)), Tuple(Mul(d, Add(a, c)), Equal(j, 4))))))),
    Variables(j, a, b, c, d),
    Assumptions(And(Element(j, Set(1, 2, 3, 4)), Element(Matrix2x2(a, b, c, d), SL2Z))))

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Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2019-09-20 18:07:53.062439 UTC