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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([483,460,126]))
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pari:[g,chi] = znchar(Mod(23,40310))
\(\chi_{40310}(23,\cdot)\)
\(\chi_{40310}(103,\cdot)\)
\(\chi_{40310}(223,\cdot)\)
\(\chi_{40310}(373,\cdot)\)
\(\chi_{40310}(703,\cdot)\)
\(\chi_{40310}(777,\cdot)\)
\(\chi_{40310}(857,\cdot)\)
\(\chi_{40310}(893,\cdot)\)
\(\chi_{40310}(1067,\cdot)\)
\(\chi_{40310}(1417,\cdot)\)
\(\chi_{40310}(1707,\cdot)\)
\(\chi_{40310}(1727,\cdot)\)
\(\chi_{40310}(1763,\cdot)\)
\(\chi_{40310}(2133,\cdot)\)
\(\chi_{40310}(2257,\cdot)\)
\(\chi_{40310}(2327,\cdot)\)
\(\chi_{40310}(2373,\cdot)\)
\(\chi_{40310}(2423,\cdot)\)
\(\chi_{40310}(2577,\cdot)\)
\(\chi_{40310}(2597,\cdot)\)
\(\chi_{40310}(2717,\cdot)\)
\(\chi_{40310}(2807,\cdot)\)
\(\chi_{40310}(2867,\cdot)\)
\(\chi_{40310}(3003,\cdot)\)
\(\chi_{40310}(3097,\cdot)\)
\(\chi_{40310}(3273,\cdot)\)
\(\chi_{40310}(3967,\cdot)\)
\(\chi_{40310}(3997,\cdot)\)
\(\chi_{40310}(4113,\cdot)\)
\(\chi_{40310}(4257,\cdot)\)
...
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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 ]
\((24187,19461,16821)\) → \((-i,e\left(\frac{5}{7}\right),e\left(\frac{9}{46}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
| \( \chi_{ 40310 }(23, a) \) |
\(1\) | \(1\) | \(e\left(\frac{543}{644}\right)\) | \(e\left(\frac{67}{644}\right)\) | \(e\left(\frac{221}{322}\right)\) | \(e\left(\frac{117}{161}\right)\) | \(e\left(\frac{405}{644}\right)\) | \(e\left(\frac{63}{92}\right)\) | \(e\left(\frac{139}{161}\right)\) | \(e\left(\frac{305}{322}\right)\) | \(e\left(\frac{527}{644}\right)\) | \(e\left(\frac{341}{644}\right)\) |
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sage:chi.jacobi_sum(n)
