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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4011, base_ring=CyclotomicField(190))
M = H._module
chi = DirichletCharacter(H, M([95,0,2]))
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pari:[g,chi] = znchar(Mod(743,4011))
\(\chi_{4011}(8,\cdot)\)
\(\chi_{4011}(50,\cdot)\)
\(\chi_{4011}(92,\cdot)\)
\(\chi_{4011}(134,\cdot)\)
\(\chi_{4011}(218,\cdot)\)
\(\chi_{4011}(239,\cdot)\)
\(\chi_{4011}(281,\cdot)\)
\(\chi_{4011}(386,\cdot)\)
\(\chi_{4011}(428,\cdot)\)
\(\chi_{4011}(449,\cdot)\)
\(\chi_{4011}(512,\cdot)\)
\(\chi_{4011}(554,\cdot)\)
\(\chi_{4011}(575,\cdot)\)
\(\chi_{4011}(596,\cdot)\)
\(\chi_{4011}(638,\cdot)\)
\(\chi_{4011}(659,\cdot)\)
\(\chi_{4011}(701,\cdot)\)
\(\chi_{4011}(722,\cdot)\)
\(\chi_{4011}(743,\cdot)\)
\(\chi_{4011}(911,\cdot)\)
\(\chi_{4011}(995,\cdot)\)
\(\chi_{4011}(1058,\cdot)\)
\(\chi_{4011}(1163,\cdot)\)
\(\chi_{4011}(1205,\cdot)\)
\(\chi_{4011}(1226,\cdot)\)
\(\chi_{4011}(1352,\cdot)\)
\(\chi_{4011}(1415,\cdot)\)
\(\chi_{4011}(1457,\cdot)\)
\(\chi_{4011}(1499,\cdot)\)
\(\chi_{4011}(1541,\cdot)\)
...
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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 ]
\((2675,2866,2311)\) → \((-1,1,e\left(\frac{1}{95}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 4011 }(743, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{183}{190}\right)\) | \(e\left(\frac{88}{95}\right)\) | \(e\left(\frac{1}{38}\right)\) | \(e\left(\frac{169}{190}\right)\) | \(e\left(\frac{94}{95}\right)\) | \(e\left(\frac{15}{38}\right)\) | \(e\left(\frac{17}{95}\right)\) | \(e\left(\frac{81}{95}\right)\) | \(e\left(\frac{101}{190}\right)\) | \(e\left(\frac{1}{95}\right)\) |
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sage:chi.jacobi_sum(n)
