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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(40310, base_ring=CyclotomicField(644))
M = H._module
chi = DirichletCharacter(H, M([322,391,280]))
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pari:[g,chi] = znchar(Mod(79,40310))
\(\chi_{40310}(79,\cdot)\)
\(\chi_{40310}(369,\cdot)\)
\(\chi_{40310}(409,\cdot)\)
\(\chi_{40310}(619,\cdot)\)
\(\chi_{40310}(739,\cdot)\)
\(\chi_{40310}(889,\cdot)\)
\(\chi_{40310}(959,\cdot)\)
\(\chi_{40310}(1469,\cdot)\)
\(\chi_{40310}(1519,\cdot)\)
\(\chi_{40310}(1609,\cdot)\)
\(\chi_{40310}(1759,\cdot)\)
\(\chi_{40310}(1859,\cdot)\)
\(\chi_{40310}(2009,\cdot)\)
\(\chi_{40310}(2119,\cdot)\)
\(\chi_{40310}(2149,\cdot)\)
\(\chi_{40310}(2289,\cdot)\)
\(\chi_{40310}(2399,\cdot)\)
\(\chi_{40310}(2479,\cdot)\)
\(\chi_{40310}(2579,\cdot)\)
\(\chi_{40310}(3019,\cdot)\)
\(\chi_{40310}(3309,\cdot)\)
\(\chi_{40310}(3519,\cdot)\)
\(\chi_{40310}(3669,\cdot)\)
\(\chi_{40310}(3739,\cdot)\)
\(\chi_{40310}(3759,\cdot)\)
\(\chi_{40310}(3789,\cdot)\)
\(\chi_{40310}(3859,\cdot)\)
\(\chi_{40310}(4249,\cdot)\)
\(\chi_{40310}(4389,\cdot)\)
\(\chi_{40310}(4539,\cdot)\)
...
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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 ]
\((24187,19461,16821)\) → \((-1,e\left(\frac{17}{28}\right),e\left(\frac{10}{23}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
| \( \chi_{ 40310 }(79, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{233}{644}\right)\) | \(e\left(\frac{169}{322}\right)\) | \(e\left(\frac{233}{322}\right)\) | \(e\left(\frac{143}{644}\right)\) | \(e\left(\frac{41}{161}\right)\) | \(e\left(\frac{71}{92}\right)\) | \(e\left(\frac{635}{644}\right)\) | \(e\left(\frac{571}{644}\right)\) | \(e\left(\frac{123}{322}\right)\) | \(e\left(\frac{55}{644}\right)\) |
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sage:chi.jacobi_sum(n)
