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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(40310, base_ring=CyclotomicField(92))
M = H._module
chi = DirichletCharacter(H, M([23,46,68]))
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pari:[g,chi] = znchar(Mod(57,40310))
\(\chi_{40310}(57,\cdot)\)
\(\chi_{40310}(173,\cdot)\)
\(\chi_{40310}(1797,\cdot)\)
\(\chi_{40310}(1913,\cdot)\)
\(\chi_{40310}(4233,\cdot)\)
\(\chi_{40310}(6727,\cdot)\)
\(\chi_{40310}(7133,\cdot)\)
\(\chi_{40310}(7307,\cdot)\)
\(\chi_{40310}(7597,\cdot)\)
\(\chi_{40310}(8003,\cdot)\)
\(\chi_{40310}(8873,\cdot)\)
\(\chi_{40310}(9627,\cdot)\)
\(\chi_{40310}(12933,\cdot)\)
\(\chi_{40310}(13687,\cdot)\)
\(\chi_{40310}(13977,\cdot)\)
\(\chi_{40310}(15137,\cdot)\)
\(\chi_{40310}(16297,\cdot)\)
\(\chi_{40310}(18037,\cdot)\)
\(\chi_{40310}(19313,\cdot)\)
\(\chi_{40310}(20357,\cdot)\)
\(\chi_{40310}(20763,\cdot)\)
\(\chi_{40310}(21053,\cdot)\)
\(\chi_{40310}(21923,\cdot)\)
\(\chi_{40310}(22213,\cdot)\)
\(\chi_{40310}(23257,\cdot)\)
\(\chi_{40310}(23953,\cdot)\)
\(\chi_{40310}(24127,\cdot)\)
\(\chi_{40310}(24243,\cdot)\)
\(\chi_{40310}(24997,\cdot)\)
\(\chi_{40310}(25983,\cdot)\)
...
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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 ]
\((24187,19461,16821)\) → \((i,-1,e\left(\frac{17}{23}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
| \( \chi_{ 40310 }(57, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{51}{92}\right)\) | \(e\left(\frac{19}{92}\right)\) | \(e\left(\frac{5}{46}\right)\) | \(e\left(\frac{31}{46}\right)\) | \(e\left(\frac{5}{92}\right)\) | \(e\left(\frac{77}{92}\right)\) | \(e\left(\frac{2}{23}\right)\) | \(e\left(\frac{35}{46}\right)\) | \(e\left(\frac{65}{92}\right)\) | \(e\left(\frac{61}{92}\right)\) |
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sage:chi.jacobi_sum(n)
