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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([196,27]))
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pari:[g,chi] = znchar(Mod(37,2349))
\(\chi_{2349}(10,\cdot)\)
\(\chi_{2349}(19,\cdot)\)
\(\chi_{2349}(37,\cdot)\)
\(\chi_{2349}(73,\cdot)\)
\(\chi_{2349}(118,\cdot)\)
\(\chi_{2349}(127,\cdot)\)
\(\chi_{2349}(172,\cdot)\)
\(\chi_{2349}(235,\cdot)\)
\(\chi_{2349}(253,\cdot)\)
\(\chi_{2349}(280,\cdot)\)
\(\chi_{2349}(316,\cdot)\)
\(\chi_{2349}(334,\cdot)\)
\(\chi_{2349}(388,\cdot)\)
\(\chi_{2349}(424,\cdot)\)
\(\chi_{2349}(478,\cdot)\)
\(\chi_{2349}(496,\cdot)\)
\(\chi_{2349}(532,\cdot)\)
\(\chi_{2349}(559,\cdot)\)
\(\chi_{2349}(577,\cdot)\)
\(\chi_{2349}(640,\cdot)\)
\(\chi_{2349}(685,\cdot)\)
\(\chi_{2349}(694,\cdot)\)
\(\chi_{2349}(739,\cdot)\)
\(\chi_{2349}(775,\cdot)\)
\(\chi_{2349}(793,\cdot)\)
\(\chi_{2349}(802,\cdot)\)
\(\chi_{2349}(820,\cdot)\)
\(\chi_{2349}(856,\cdot)\)
\(\chi_{2349}(901,\cdot)\)
\(\chi_{2349}(910,\cdot)\)
...
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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 ]
\((407,1945)\) → \((e\left(\frac{7}{9}\right),e\left(\frac{3}{28}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 2349 }(37, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{223}{252}\right)\) | \(e\left(\frac{97}{126}\right)\) | \(e\left(\frac{31}{126}\right)\) | \(e\left(\frac{46}{63}\right)\) | \(e\left(\frac{55}{84}\right)\) | \(e\left(\frac{11}{84}\right)\) | \(e\left(\frac{199}{252}\right)\) | \(e\left(\frac{19}{126}\right)\) | \(e\left(\frac{155}{252}\right)\) | \(e\left(\frac{34}{63}\right)\) |
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sage:chi.jacobi_sum(n)
