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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(305760, base_ring=CyclotomicField(168))
M = H._module
chi = DirichletCharacter(H, M([84,105,0,84,60,112]))
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pari:[g,chi] = znchar(Mod(139,305760))
\(\chi_{305760}(139,\cdot)\)
\(\chi_{305760}(7699,\cdot)\)
\(\chi_{305760}(11059,\cdot)\)
\(\chi_{305760}(21979,\cdot)\)
\(\chi_{305760}(29539,\cdot)\)
\(\chi_{305760}(32899,\cdot)\)
\(\chi_{305760}(40459,\cdot)\)
\(\chi_{305760}(43819,\cdot)\)
\(\chi_{305760}(51379,\cdot)\)
\(\chi_{305760}(54739,\cdot)\)
\(\chi_{305760}(62299,\cdot)\)
\(\chi_{305760}(73219,\cdot)\)
\(\chi_{305760}(76579,\cdot)\)
\(\chi_{305760}(84139,\cdot)\)
\(\chi_{305760}(87499,\cdot)\)
\(\chi_{305760}(98419,\cdot)\)
\(\chi_{305760}(105979,\cdot)\)
\(\chi_{305760}(109339,\cdot)\)
\(\chi_{305760}(116899,\cdot)\)
\(\chi_{305760}(120259,\cdot)\)
\(\chi_{305760}(127819,\cdot)\)
\(\chi_{305760}(131179,\cdot)\)
\(\chi_{305760}(138739,\cdot)\)
\(\chi_{305760}(149659,\cdot)\)
\(\chi_{305760}(153019,\cdot)\)
\(\chi_{305760}(160579,\cdot)\)
\(\chi_{305760}(163939,\cdot)\)
\(\chi_{305760}(174859,\cdot)\)
\(\chi_{305760}(182419,\cdot)\)
\(\chi_{305760}(185779,\cdot)\)
...
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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 ]
\((95551,114661,101921,183457,18721,211681)\) → \((-1,e\left(\frac{5}{8}\right),1,-1,e\left(\frac{5}{14}\right),e\left(\frac{2}{3}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(11\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) | \(47\) |
| \( \chi_{ 305760 }(139, a) \) |
\(1\) | \(1\) | \(e\left(\frac{97}{168}\right)\) | \(e\left(\frac{11}{42}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{83}{84}\right)\) | \(e\left(\frac{163}{168}\right)\) | \(1\) | \(e\left(\frac{37}{168}\right)\) | \(e\left(\frac{65}{84}\right)\) | \(e\left(\frac{101}{168}\right)\) | \(e\left(\frac{2}{7}\right)\) |
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sage:chi.jacobi_sum(n)
