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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([161,345,574]))
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pari:[g,chi] = znchar(Mod(27,40310))
\(\chi_{40310}(27,\cdot)\)
\(\chi_{40310}(317,\cdot)\)
\(\chi_{40310}(337,\cdot)\)
\(\chi_{40310}(583,\cdot)\)
\(\chi_{40310}(867,\cdot)\)
\(\chi_{40310}(873,\cdot)\)
\(\chi_{40310}(967,\cdot)\)
\(\chi_{40310}(1187,\cdot)\)
\(\chi_{40310}(1207,\cdot)\)
\(\chi_{40310}(1413,\cdot)\)
\(\chi_{40310}(1423,\cdot)\)
\(\chi_{40310}(1477,\cdot)\)
\(\chi_{40310}(1493,\cdot)\)
\(\chi_{40310}(1743,\cdot)\)
\(\chi_{40310}(2033,\cdot)\)
\(\chi_{40310}(2283,\cdot)\)
\(\chi_{40310}(2357,\cdot)\)
\(\chi_{40310}(2607,\cdot)\)
\(\chi_{40310}(2803,\cdot)\)
\(\chi_{40310}(2927,\cdot)\)
\(\chi_{40310}(3153,\cdot)\)
\(\chi_{40310}(3163,\cdot)\)
\(\chi_{40310}(3483,\cdot)\)
\(\chi_{40310}(3523,\cdot)\)
\(\chi_{40310}(3673,\cdot)\)
\(\chi_{40310}(3767,\cdot)\)
\(\chi_{40310}(3813,\cdot)\)
\(\chi_{40310}(4107,\cdot)\)
\(\chi_{40310}(4303,\cdot)\)
\(\chi_{40310}(4323,\cdot)\)
...
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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 ]
\((24187,19461,16821)\) → \((i,e\left(\frac{15}{28}\right),e\left(\frac{41}{46}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
| \( \chi_{ 40310 }(27, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{313}{322}\right)\) | \(e\left(\frac{157}{644}\right)\) | \(e\left(\frac{152}{161}\right)\) | \(e\left(\frac{85}{644}\right)\) | \(e\left(\frac{281}{644}\right)\) | \(e\left(\frac{20}{23}\right)\) | \(e\left(\frac{445}{644}\right)\) | \(e\left(\frac{139}{644}\right)\) | \(e\left(\frac{341}{644}\right)\) | \(e\left(\frac{295}{322}\right)\) |
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sage:chi.jacobi_sum(n)
