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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2349, base_ring=CyclotomicField(252))
M = H._module
chi = DirichletCharacter(H, M([238,81]))
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pari:[g,chi] = znchar(Mod(2339,2349))
\(\chi_{2349}(8,\cdot)\)
\(\chi_{2349}(44,\cdot)\)
\(\chi_{2349}(89,\cdot)\)
\(\chi_{2349}(98,\cdot)\)
\(\chi_{2349}(143,\cdot)\)
\(\chi_{2349}(206,\cdot)\)
\(\chi_{2349}(224,\cdot)\)
\(\chi_{2349}(251,\cdot)\)
\(\chi_{2349}(287,\cdot)\)
\(\chi_{2349}(305,\cdot)\)
\(\chi_{2349}(359,\cdot)\)
\(\chi_{2349}(395,\cdot)\)
\(\chi_{2349}(449,\cdot)\)
\(\chi_{2349}(467,\cdot)\)
\(\chi_{2349}(503,\cdot)\)
\(\chi_{2349}(530,\cdot)\)
\(\chi_{2349}(548,\cdot)\)
\(\chi_{2349}(611,\cdot)\)
\(\chi_{2349}(656,\cdot)\)
\(\chi_{2349}(665,\cdot)\)
\(\chi_{2349}(710,\cdot)\)
\(\chi_{2349}(746,\cdot)\)
\(\chi_{2349}(764,\cdot)\)
\(\chi_{2349}(773,\cdot)\)
\(\chi_{2349}(791,\cdot)\)
\(\chi_{2349}(827,\cdot)\)
\(\chi_{2349}(872,\cdot)\)
\(\chi_{2349}(881,\cdot)\)
\(\chi_{2349}(926,\cdot)\)
\(\chi_{2349}(989,\cdot)\)
...
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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 ]
\((407,1945)\) → \((e\left(\frac{17}{18}\right),e\left(\frac{9}{28}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 2349 }(2339, a) \) |
\(1\) | \(1\) | \(e\left(\frac{67}{252}\right)\) | \(e\left(\frac{67}{126}\right)\) | \(e\left(\frac{50}{63}\right)\) | \(e\left(\frac{61}{63}\right)\) | \(e\left(\frac{67}{84}\right)\) | \(e\left(\frac{5}{84}\right)\) | \(e\left(\frac{79}{252}\right)\) | \(e\left(\frac{43}{126}\right)\) | \(e\left(\frac{59}{252}\right)\) | \(e\left(\frac{4}{63}\right)\) |
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sage:chi.jacobi_sum(n)
