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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(28665, base_ring=CyclotomicField(84))
M = H._module
chi = DirichletCharacter(H, M([28,63,6,0]))
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pari:[g,chi] = znchar(Mod(27418,28665))
\(\chi_{28665}(1483,\cdot)\)
\(\chi_{28665}(2848,\cdot)\)
\(\chi_{28665}(3667,\cdot)\)
\(\chi_{28665}(5578,\cdot)\)
\(\chi_{28665}(6397,\cdot)\)
\(\chi_{28665}(6943,\cdot)\)
\(\chi_{28665}(7762,\cdot)\)
\(\chi_{28665}(9673,\cdot)\)
\(\chi_{28665}(10492,\cdot)\)
\(\chi_{28665}(11038,\cdot)\)
\(\chi_{28665}(14587,\cdot)\)
\(\chi_{28665}(15133,\cdot)\)
\(\chi_{28665}(15952,\cdot)\)
\(\chi_{28665}(17863,\cdot)\)
\(\chi_{28665}(18682,\cdot)\)
\(\chi_{28665}(19228,\cdot)\)
\(\chi_{28665}(20047,\cdot)\)
\(\chi_{28665}(21958,\cdot)\)
\(\chi_{28665}(22777,\cdot)\)
\(\chi_{28665}(24142,\cdot)\)
\(\chi_{28665}(26053,\cdot)\)
\(\chi_{28665}(26872,\cdot)\)
\(\chi_{28665}(27418,\cdot)\)
\(\chi_{28665}(28237,\cdot)\)
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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 ]
\((25481,11467,18721,11026)\) → \((e\left(\frac{1}{3}\right),-i,e\left(\frac{1}{14}\right),1)\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(8\) | \(11\) | \(16\) | \(17\) | \(19\) | \(22\) | \(23\) | \(29\) |
| \( \chi_{ 28665 }(27418, a) \) |
\(1\) | \(1\) | \(e\left(\frac{79}{84}\right)\) | \(e\left(\frac{37}{42}\right)\) | \(e\left(\frac{23}{28}\right)\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{15}{28}\right)\) | \(1\) | \(e\left(\frac{11}{84}\right)\) | \(e\left(\frac{53}{84}\right)\) | \(e\left(\frac{5}{42}\right)\) |
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sage:chi.jacobi_sum(n)
