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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4320, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([18,27,16,27]))
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pari:[g,chi] = znchar(Mod(823,4320))
\(\chi_{4320}(7,\cdot)\)
\(\chi_{4320}(823,\cdot)\)
\(\chi_{4320}(967,\cdot)\)
\(\chi_{4320}(1303,\cdot)\)
\(\chi_{4320}(1447,\cdot)\)
\(\chi_{4320}(2263,\cdot)\)
\(\chi_{4320}(2407,\cdot)\)
\(\chi_{4320}(2743,\cdot)\)
\(\chi_{4320}(2887,\cdot)\)
\(\chi_{4320}(3703,\cdot)\)
\(\chi_{4320}(3847,\cdot)\)
\(\chi_{4320}(4183,\cdot)\)
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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 ]
\((2431,3781,2081,3457)\) → \((-1,-i,e\left(\frac{4}{9}\right),-i)\)
| \(a\) |
\(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 4320 }(823, a) \) |
\(1\) | \(1\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{1}{36}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{5}{36}\right)\) | \(e\left(\frac{7}{36}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{18}\right)\) |
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sage:chi.jacobi_sum(n)
