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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1682, base_ring=CyclotomicField(58))
M = H._module
chi = DirichletCharacter(H, M([5]))
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pari:[g,chi] = znchar(Mod(347,1682))
\(\chi_{1682}(57,\cdot)\)
\(\chi_{1682}(115,\cdot)\)
\(\chi_{1682}(173,\cdot)\)
\(\chi_{1682}(231,\cdot)\)
\(\chi_{1682}(289,\cdot)\)
\(\chi_{1682}(347,\cdot)\)
\(\chi_{1682}(405,\cdot)\)
\(\chi_{1682}(463,\cdot)\)
\(\chi_{1682}(521,\cdot)\)
\(\chi_{1682}(579,\cdot)\)
\(\chi_{1682}(637,\cdot)\)
\(\chi_{1682}(695,\cdot)\)
\(\chi_{1682}(753,\cdot)\)
\(\chi_{1682}(811,\cdot)\)
\(\chi_{1682}(869,\cdot)\)
\(\chi_{1682}(927,\cdot)\)
\(\chi_{1682}(985,\cdot)\)
\(\chi_{1682}(1043,\cdot)\)
\(\chi_{1682}(1101,\cdot)\)
\(\chi_{1682}(1159,\cdot)\)
\(\chi_{1682}(1217,\cdot)\)
\(\chi_{1682}(1275,\cdot)\)
\(\chi_{1682}(1333,\cdot)\)
\(\chi_{1682}(1391,\cdot)\)
\(\chi_{1682}(1449,\cdot)\)
\(\chi_{1682}(1507,\cdot)\)
\(\chi_{1682}(1565,\cdot)\)
\(\chi_{1682}(1623,\cdot)\)
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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 ]
\(843\) → \(e\left(\frac{5}{58}\right)\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 1682 }(347, a) \) |
\(1\) | \(1\) | \(e\left(\frac{17}{58}\right)\) | \(e\left(\frac{1}{29}\right)\) | \(e\left(\frac{12}{29}\right)\) | \(e\left(\frac{17}{29}\right)\) | \(e\left(\frac{57}{58}\right)\) | \(e\left(\frac{2}{29}\right)\) | \(e\left(\frac{19}{58}\right)\) | \(e\left(\frac{25}{58}\right)\) | \(e\left(\frac{51}{58}\right)\) | \(e\left(\frac{41}{58}\right)\) |
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sage:chi.jacobi_sum(n)
