Fungrim entry: fc3ef5

{\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 \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
`Pow`	${a}^{b}$	Power
`JacobiThetaEpsilon`	$\varepsilon_{j}\!\left(a, b, c, d\right)$	Root of unity in modular transformation of Jacobi theta functions
`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("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))))

Topics using this entry

Lattice transformations for Jacobi theta functions