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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([33,33,22,3]))
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pari:[g,chi] = znchar(Mod(4559,4830))
\(\chi_{4830}(149,\cdot)\)
\(\chi_{4830}(359,\cdot)\)
\(\chi_{4830}(389,\cdot)\)
\(\chi_{4830}(569,\cdot)\)
\(\chi_{4830}(779,\cdot)\)
\(\chi_{4830}(1019,\cdot)\)
\(\chi_{4830}(1229,\cdot)\)
\(\chi_{4830}(1859,\cdot)\)
\(\chi_{4830}(2039,\cdot)\)
\(\chi_{4830}(2459,\cdot)\)
\(\chi_{4830}(2489,\cdot)\)
\(\chi_{4830}(2909,\cdot)\)
\(\chi_{4830}(3089,\cdot)\)
\(\chi_{4830}(3119,\cdot)\)
\(\chi_{4830}(3299,\cdot)\)
\(\chi_{4830}(3329,\cdot)\)
\(\chi_{4830}(3539,\cdot)\)
\(\chi_{4830}(3929,\cdot)\)
\(\chi_{4830}(4559,\cdot)\)
\(\chi_{4830}(4799,\cdot)\)
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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 ]
\((3221,967,2761,1891)\) → \((-1,-1,e\left(\frac{1}{3}\right),e\left(\frac{1}{22}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(11\) | \(13\) | \(17\) | \(19\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) | \(47\) |
| \( \chi_{ 4830 }(4559, a) \) |
\(1\) | \(1\) | \(e\left(\frac{8}{33}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{43}{66}\right)\) | \(e\left(\frac{23}{66}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{20}{33}\right)\) | \(e\left(\frac{4}{33}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{2}{3}\right)\) |
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sage:chi.jacobi_sum(n)
