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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4334, base_ring=CyclotomicField(196))
M = H._module
chi = DirichletCharacter(H, M([0,171]))
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pari:[g,chi] = znchar(Mod(485,4334))
\(\chi_{4334}(45,\cdot)\)
\(\chi_{4334}(67,\cdot)\)
\(\chi_{4334}(89,\cdot)\)
\(\chi_{4334}(111,\cdot)\)
\(\chi_{4334}(199,\cdot)\)
\(\chi_{4334}(243,\cdot)\)
\(\chi_{4334}(397,\cdot)\)
\(\chi_{4334}(485,\cdot)\)
\(\chi_{4334}(573,\cdot)\)
\(\chi_{4334}(639,\cdot)\)
\(\chi_{4334}(771,\cdot)\)
\(\chi_{4334}(793,\cdot)\)
\(\chi_{4334}(815,\cdot)\)
\(\chi_{4334}(859,\cdot)\)
\(\chi_{4334}(903,\cdot)\)
\(\chi_{4334}(947,\cdot)\)
\(\chi_{4334}(1035,\cdot)\)
\(\chi_{4334}(1057,\cdot)\)
\(\chi_{4334}(1079,\cdot)\)
\(\chi_{4334}(1255,\cdot)\)
\(\chi_{4334}(1277,\cdot)\)
\(\chi_{4334}(1299,\cdot)\)
\(\chi_{4334}(1321,\cdot)\)
\(\chi_{4334}(1387,\cdot)\)
\(\chi_{4334}(1409,\cdot)\)
\(\chi_{4334}(1431,\cdot)\)
\(\chi_{4334}(1453,\cdot)\)
\(\chi_{4334}(1497,\cdot)\)
\(\chi_{4334}(1519,\cdot)\)
\(\chi_{4334}(1541,\cdot)\)
...
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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 ]
\((1971,199)\) → \((1,e\left(\frac{171}{196}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) | \(23\) |
| \( \chi_{ 4334 }(485, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{179}{196}\right)\) | \(e\left(\frac{127}{196}\right)\) | \(e\left(\frac{37}{98}\right)\) | \(e\left(\frac{81}{98}\right)\) | \(e\left(\frac{159}{196}\right)\) | \(e\left(\frac{55}{98}\right)\) | \(e\left(\frac{141}{196}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{57}{196}\right)\) | \(e\left(\frac{34}{49}\right)\) |
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sage:chi.jacobi_sum(n)
