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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(640332, base_ring=CyclotomicField(6930))
M = H._module
chi = DirichletCharacter(H, M([0,770,3300,6174]))
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pari:[g,chi] = znchar(Mod(10453,640332))
\(\chi_{640332}(25,\cdot)\)
\(\chi_{640332}(625,\cdot)\)
\(\chi_{640332}(1285,\cdot)\)
\(\chi_{640332}(1633,\cdot)\)
\(\chi_{640332}(1885,\cdot)\)
\(\chi_{640332}(2293,\cdot)\)
\(\chi_{640332}(2545,\cdot)\)
\(\chi_{640332}(2797,\cdot)\)
\(\chi_{640332}(4057,\cdot)\)
\(\chi_{640332}(4405,\cdot)\)
\(\chi_{640332}(4657,\cdot)\)
\(\chi_{640332}(4909,\cdot)\)
\(\chi_{640332}(5317,\cdot)\)
\(\chi_{640332}(6169,\cdot)\)
\(\chi_{640332}(7177,\cdot)\)
\(\chi_{640332}(7681,\cdot)\)
\(\chi_{640332}(7837,\cdot)\)
\(\chi_{640332}(8089,\cdot)\)
\(\chi_{640332}(8341,\cdot)\)
\(\chi_{640332}(8941,\cdot)\)
\(\chi_{640332}(9601,\cdot)\)
\(\chi_{640332}(10201,\cdot)\)
\(\chi_{640332}(10453,\cdot)\)
\(\chi_{640332}(10609,\cdot)\)
\(\chi_{640332}(10861,\cdot)\)
\(\chi_{640332}(11113,\cdot)\)
\(\chi_{640332}(11713,\cdot)\)
\(\chi_{640332}(12373,\cdot)\)
\(\chi_{640332}(12973,\cdot)\)
\(\chi_{640332}(13225,\cdot)\)
...
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sage:chi.galois_orbit()
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pari:order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
\((320167,450605,339769,179929)\) → \((1,e\left(\frac{1}{9}\right),e\left(\frac{10}{21}\right),e\left(\frac{49}{55}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 640332 }(10453, a) \) |
\(1\) | \(1\) | \(e\left(\frac{1013}{3465}\right)\) | \(e\left(\frac{2027}{3465}\right)\) | \(e\left(\frac{87}{385}\right)\) | \(e\left(\frac{52}{55}\right)\) | \(e\left(\frac{472}{693}\right)\) | \(e\left(\frac{2026}{3465}\right)\) | \(e\left(\frac{2869}{3465}\right)\) | \(e\left(\frac{86}{495}\right)\) | \(e\left(\frac{373}{1155}\right)\) | \(e\left(\frac{1811}{3465}\right)\) |
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sage:chi.jacobi_sum(n)
