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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(12138, base_ring=CyclotomicField(816))
M = H._module
chi = DirichletCharacter(H, M([0,680,309]))
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pari:[g,chi] = znchar(Mod(691,12138))
\(\chi_{12138}(31,\cdot)\)
\(\chi_{12138}(61,\cdot)\)
\(\chi_{12138}(73,\cdot)\)
\(\chi_{12138}(199,\cdot)\)
\(\chi_{12138}(241,\cdot)\)
\(\chi_{12138}(283,\cdot)\)
\(\chi_{12138}(313,\cdot)\)
\(\chi_{12138}(367,\cdot)\)
\(\chi_{12138}(397,\cdot)\)
\(\chi_{12138}(439,\cdot)\)
\(\chi_{12138}(481,\cdot)\)
\(\chi_{12138}(607,\cdot)\)
\(\chi_{12138}(619,\cdot)\)
\(\chi_{12138}(649,\cdot)\)
\(\chi_{12138}(691,\cdot)\)
\(\chi_{12138}(703,\cdot)\)
\(\chi_{12138}(745,\cdot)\)
\(\chi_{12138}(775,\cdot)\)
\(\chi_{12138}(787,\cdot)\)
\(\chi_{12138}(913,\cdot)\)
\(\chi_{12138}(955,\cdot)\)
\(\chi_{12138}(997,\cdot)\)
\(\chi_{12138}(1027,\cdot)\)
\(\chi_{12138}(1111,\cdot)\)
\(\chi_{12138}(1153,\cdot)\)
\(\chi_{12138}(1195,\cdot)\)
\(\chi_{12138}(1321,\cdot)\)
\(\chi_{12138}(1333,\cdot)\)
\(\chi_{12138}(1363,\cdot)\)
\(\chi_{12138}(1417,\cdot)\)
...
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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 ]
\((8093,10405,9829)\) → \((1,e\left(\frac{5}{6}\right),e\left(\frac{103}{272}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 12138 }(691, a) \) |
\(1\) | \(1\) | \(e\left(\frac{721}{816}\right)\) | \(e\left(\frac{35}{816}\right)\) | \(e\left(\frac{49}{68}\right)\) | \(e\left(\frac{191}{408}\right)\) | \(e\left(\frac{91}{816}\right)\) | \(e\left(\frac{313}{408}\right)\) | \(e\left(\frac{91}{272}\right)\) | \(e\left(\frac{197}{816}\right)\) | \(e\left(\frac{421}{816}\right)\) | \(e\left(\frac{69}{272}\right)\) |
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sage:chi.jacobi_sum(n)
