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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([0,33,11,3]))
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pari:[g,chi] = znchar(Mod(2719,4830))
\(\chi_{4830}(19,\cdot)\)
\(\chi_{4830}(199,\cdot)\)
\(\chi_{4830}(619,\cdot)\)
\(\chi_{4830}(649,\cdot)\)
\(\chi_{4830}(1069,\cdot)\)
\(\chi_{4830}(1249,\cdot)\)
\(\chi_{4830}(1279,\cdot)\)
\(\chi_{4830}(1459,\cdot)\)
\(\chi_{4830}(1489,\cdot)\)
\(\chi_{4830}(1699,\cdot)\)
\(\chi_{4830}(2089,\cdot)\)
\(\chi_{4830}(2719,\cdot)\)
\(\chi_{4830}(2959,\cdot)\)
\(\chi_{4830}(3139,\cdot)\)
\(\chi_{4830}(3349,\cdot)\)
\(\chi_{4830}(3379,\cdot)\)
\(\chi_{4830}(3559,\cdot)\)
\(\chi_{4830}(3769,\cdot)\)
\(\chi_{4830}(4009,\cdot)\)
\(\chi_{4830}(4219,\cdot)\)
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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 ]
\((3221,967,2761,1891)\) → \((1,-1,e\left(\frac{1}{6}\right),e\left(\frac{1}{22}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(11\) | \(13\) | \(17\) | \(19\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) | \(47\) |
| \( \chi_{ 4830 }(2719, a) \) |
\(1\) | \(1\) | \(e\left(\frac{5}{66}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{65}{66}\right)\) | \(e\left(\frac{17}{33}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{29}{66}\right)\) | \(e\left(\frac{26}{33}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{1}{3}\right)\) |
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sage:chi.jacobi_sum(n)
