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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2667, base_ring=CyclotomicField(126))
M = H._module
chi = DirichletCharacter(H, M([0,0,71]))
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pari:[g,chi] = znchar(Mod(43,2667))
\(\chi_{2667}(43,\cdot)\)
\(\chi_{2667}(85,\cdot)\)
\(\chi_{2667}(106,\cdot)\)
\(\chi_{2667}(337,\cdot)\)
\(\chi_{2667}(547,\cdot)\)
\(\chi_{2667}(736,\cdot)\)
\(\chi_{2667}(820,\cdot)\)
\(\chi_{2667}(946,\cdot)\)
\(\chi_{2667}(967,\cdot)\)
\(\chi_{2667}(1030,\cdot)\)
\(\chi_{2667}(1072,\cdot)\)
\(\chi_{2667}(1198,\cdot)\)
\(\chi_{2667}(1240,\cdot)\)
\(\chi_{2667}(1261,\cdot)\)
\(\chi_{2667}(1282,\cdot)\)
\(\chi_{2667}(1366,\cdot)\)
\(\chi_{2667}(1450,\cdot)\)
\(\chi_{2667}(1513,\cdot)\)
\(\chi_{2667}(1744,\cdot)\)
\(\chi_{2667}(1765,\cdot)\)
\(\chi_{2667}(1807,\cdot)\)
\(\chi_{2667}(1870,\cdot)\)
\(\chi_{2667}(1912,\cdot)\)
\(\chi_{2667}(1996,\cdot)\)
\(\chi_{2667}(2017,\cdot)\)
\(\chi_{2667}(2038,\cdot)\)
\(\chi_{2667}(2080,\cdot)\)
\(\chi_{2667}(2269,\cdot)\)
\(\chi_{2667}(2332,\cdot)\)
\(\chi_{2667}(2353,\cdot)\)
...
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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 ]
\((890,1144,2416)\) → \((1,1,e\left(\frac{71}{126}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 2667 }(43, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{25}{42}\right)\) | \(e\left(\frac{20}{63}\right)\) | \(e\left(\frac{61}{63}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{26}{63}\right)\) | \(e\left(\frac{1}{3}\right)\) |
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sage:chi.jacobi_sum(n)
