Fungrim entry: 03fbe8

\chi_{q \, . \, \ell}\!\left(n\right) = \exp\!\left(2 \pi i \left(\frac{\left(1 - x\right) \left(1 - y\right)}{8} + \frac{a b}{{2}^{e - 2}}\right)\right)\; \text{ where } q = {2}^{e},\;L(k) = \begin{cases} \left(1, \log_{5}\!\left(k\right) \bmod q\right), & k \in \left\{ {5}^{i} \bmod q : i \in \mathbb{Z}_{\ge 1} \right\}\\\left(-1, \log_{5}\!\left(-k\right) \bmod q\right), & k \in \left\{ -{5}^{i} \bmod q : i \in \mathbb{Z}_{\ge 1} \right\}\\ \end{cases},\;\left(x, a\right) = L(\ell),\;\left(y, b\right) = L(n)

Assumptions:

e \in \mathbb{Z}_{\ge 3} \;\mathbin{\operatorname{and}}\; \ell \in \{1, 2, \ldots, {2}^{e} - 1\} \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; \gcd\!\left(\ell, 2\right) = \gcd\!\left(n, 2\right) = 1

TeX:

\chi_{q \, . \, \ell}\!\left(n\right) = \exp\!\left(2 \pi i \left(\frac{\left(1 - x\right) \left(1 - y\right)}{8} + \frac{a b}{{2}^{e - 2}}\right)\right)\; \text{ where } q = {2}^{e},\;L(k) = \begin{cases} \left(1, \log_{5}\!\left(k\right) \bmod q\right), & k \in \left\{ {5}^{i} \bmod q : i \in \mathbb{Z}_{\ge 1} \right\}\\\left(-1, \log_{5}\!\left(-k\right) \bmod q\right), & k \in \left\{ -{5}^{i} \bmod q : i \in \mathbb{Z}_{\ge 1} \right\}\\ \end{cases},\;\left(x, a\right) = L(\ell),\;\left(y, b\right) = L(n)

e \in \mathbb{Z}_{\ge 3} \;\mathbin{\operatorname{and}}\; \ell \in \{1, 2, \ldots, {2}^{e} - 1\} \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; \gcd\!\left(\ell, 2\right) = \gcd\!\left(n, 2\right) = 1

Definitions:

Fungrim symbol	Notation	Short description
`DirichletCharacter`	$\chi_{q \, . \, \ell}$	Dirichlet character
`Exp`	${e}^{z}$	Exponential function
`Pi`	$\pi$	The constant pi (3.14...)
`ConstI`	$i$	Imaginary unit
`Pow`	${a}^{b}$	Power
`DiscreteLog`	$\log_{b}\!\left(x\right) \bmod q$	Discrete logarithm
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`Range`	$\{a, a + 1, \ldots, b\}$	Integers between given endpoints
`ZZ`	$\mathbb{Z}$	Integers
`GCD`	$\gcd\!\left(a, b\right)$	Greatest common divisor

Source code for this entry:

Entry(ID("03fbe8"),
    Formula(Where(Equal(DirichletCharacter(q, ell, n), Exp(Mul(Mul(Mul(2, Pi), ConstI), Add(Div(Mul(Sub(1, x), Sub(1, y)), 8), Div(Mul(a, b), Pow(2, Sub(e, 2))))))), Equal(q, Pow(2, e)), Equal(L(k), Cases(Tuple(Tuple(1, DiscreteLog(k, 5, q)), Element(k, Set(Mod(Pow(5, i), q), For(i), Element(i, ZZGreaterEqual(1))))), Tuple(Tuple(-1, DiscreteLog(Neg(k), 5, q)), Element(k, Set(Mod(Neg(Pow(5, i)), q), For(i), Element(i, ZZGreaterEqual(1))))))), Equal(Tuple(x, a), L(ell)), Equal(Tuple(y, b), L(n)))),
    Variables(e, ell, n),
    Assumptions(And(Element(e, ZZGreaterEqual(3)), Element(ell, Range(1, Sub(Pow(2, e), 1))), Element(n, ZZ), Equal(GCD(ell, 2), GCD(n, 2), 1))))

Topics using this entry

Dirichlet characters