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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4830, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([0,33,22,12]))
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pari:[g,chi] = znchar(Mod(4489,4830))
\(\chi_{4830}(289,\cdot)\)
\(\chi_{4830}(499,\cdot)\)
\(\chi_{4830}(739,\cdot)\)
\(\chi_{4830}(949,\cdot)\)
\(\chi_{4830}(1129,\cdot)\)
\(\chi_{4830}(1159,\cdot)\)
\(\chi_{4830}(1369,\cdot)\)
\(\chi_{4830}(1549,\cdot)\)
\(\chi_{4830}(1789,\cdot)\)
\(\chi_{4830}(2419,\cdot)\)
\(\chi_{4830}(2809,\cdot)\)
\(\chi_{4830}(3019,\cdot)\)
\(\chi_{4830}(3049,\cdot)\)
\(\chi_{4830}(3229,\cdot)\)
\(\chi_{4830}(3259,\cdot)\)
\(\chi_{4830}(3439,\cdot)\)
\(\chi_{4830}(3859,\cdot)\)
\(\chi_{4830}(3889,\cdot)\)
\(\chi_{4830}(4309,\cdot)\)
\(\chi_{4830}(4489,\cdot)\)
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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 ]
\((3221,967,2761,1891)\) → \((1,-1,e\left(\frac{1}{3}\right),e\left(\frac{2}{11}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(11\) | \(13\) | \(17\) | \(19\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) | \(47\) |
| \( \chi_{ 4830 }(4489, a) \) |
\(1\) | \(1\) | \(e\left(\frac{32}{33}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{7}{66}\right)\) | \(e\left(\frac{13}{33}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{14}{33}\right)\) | \(e\left(\frac{65}{66}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{1}{6}\right)\) |
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sage:chi.jacobi_sum(n)
