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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4998, base_ring=CyclotomicField(336))
M = H._module
chi = DirichletCharacter(H, M([0,40,273]))
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pari:[g,chi] = znchar(Mod(1321,4998))
\(\chi_{4998}(61,\cdot)\)
\(\chi_{4998}(73,\cdot)\)
\(\chi_{4998}(199,\cdot)\)
\(\chi_{4998}(241,\cdot)\)
\(\chi_{4998}(283,\cdot)\)
\(\chi_{4998}(367,\cdot)\)
\(\chi_{4998}(397,\cdot)\)
\(\chi_{4998}(439,\cdot)\)
\(\chi_{4998}(481,\cdot)\)
\(\chi_{4998}(649,\cdot)\)
\(\chi_{4998}(691,\cdot)\)
\(\chi_{4998}(703,\cdot)\)
\(\chi_{4998}(745,\cdot)\)
\(\chi_{4998}(775,\cdot)\)
\(\chi_{4998}(787,\cdot)\)
\(\chi_{4998}(955,\cdot)\)
\(\chi_{4998}(997,\cdot)\)
\(\chi_{4998}(1027,\cdot)\)
\(\chi_{4998}(1081,\cdot)\)
\(\chi_{4998}(1111,\cdot)\)
\(\chi_{4998}(1153,\cdot)\)
\(\chi_{4998}(1321,\cdot)\)
\(\chi_{4998}(1333,\cdot)\)
\(\chi_{4998}(1363,\cdot)\)
\(\chi_{4998}(1405,\cdot)\)
\(\chi_{4998}(1417,\cdot)\)
\(\chi_{4998}(1459,\cdot)\)
\(\chi_{4998}(1627,\cdot)\)
\(\chi_{4998}(1669,\cdot)\)
\(\chi_{4998}(1711,\cdot)\)
...
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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 ]
\((1667,2551,4117)\) → \((1,e\left(\frac{5}{42}\right),e\left(\frac{13}{16}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 4998 }(1321, a) \) |
\(1\) | \(1\) | \(e\left(\frac{173}{336}\right)\) | \(e\left(\frac{151}{336}\right)\) | \(e\left(\frac{5}{28}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{239}{336}\right)\) | \(e\left(\frac{5}{168}\right)\) | \(e\left(\frac{79}{112}\right)\) | \(e\left(\frac{7}{48}\right)\) | \(e\left(\frac{209}{336}\right)\) | \(e\left(\frac{81}{112}\right)\) |
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sage:chi.jacobi_sum(n)
