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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7742, base_ring=CyclotomicField(546))
M = H._module
chi = DirichletCharacter(H, M([208,504]))
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pari:[g,chi] = znchar(Mod(417,7742))
\(\chi_{7742}(65,\cdot)\)
\(\chi_{7742}(179,\cdot)\)
\(\chi_{7742}(247,\cdot)\)
\(\chi_{7742}(289,\cdot)\)
\(\chi_{7742}(403,\cdot)\)
\(\chi_{7742}(417,\cdot)\)
\(\chi_{7742}(457,\cdot)\)
\(\chi_{7742}(541,\cdot)\)
\(\chi_{7742}(571,\cdot)\)
\(\chi_{7742}(599,\cdot)\)
\(\chi_{7742}(653,\cdot)\)
\(\chi_{7742}(697,\cdot)\)
\(\chi_{7742}(877,\cdot)\)
\(\chi_{7742}(879,\cdot)\)
\(\chi_{7742}(891,\cdot)\)
\(\chi_{7742}(907,\cdot)\)
\(\chi_{7742}(921,\cdot)\)
\(\chi_{7742}(933,\cdot)\)
\(\chi_{7742}(1045,\cdot)\)
\(\chi_{7742}(1073,\cdot)\)
\(\chi_{7742}(1089,\cdot)\)
\(\chi_{7742}(1171,\cdot)\)
\(\chi_{7742}(1173,\cdot)\)
\(\chi_{7742}(1285,\cdot)\)
\(\chi_{7742}(1381,\cdot)\)
\(\chi_{7742}(1395,\cdot)\)
\(\chi_{7742}(1509,\cdot)\)
\(\chi_{7742}(1523,\cdot)\)
\(\chi_{7742}(1563,\cdot)\)
\(\chi_{7742}(1565,\cdot)\)
...
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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 ]
\((2845,4901)\) → \((e\left(\frac{8}{21}\right),e\left(\frac{12}{13}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) |
| \( \chi_{ 7742 }(417, a) \) |
\(1\) | \(1\) | \(e\left(\frac{83}{273}\right)\) | \(e\left(\frac{76}{273}\right)\) | \(e\left(\frac{166}{273}\right)\) | \(e\left(\frac{2}{273}\right)\) | \(e\left(\frac{87}{91}\right)\) | \(e\left(\frac{53}{91}\right)\) | \(e\left(\frac{248}{273}\right)\) | \(e\left(\frac{34}{39}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{152}{273}\right)\) |
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sage:chi.jacobi_sum(n)
