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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4320, base_ring=CyclotomicField(24))
M = H._module
chi = DirichletCharacter(H, M([12,3,8,0]))
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pari:[g,chi] = znchar(Mod(91,4320))
\(\chi_{4320}(91,\cdot)\)
\(\chi_{4320}(451,\cdot)\)
\(\chi_{4320}(1171,\cdot)\)
\(\chi_{4320}(1531,\cdot)\)
\(\chi_{4320}(2251,\cdot)\)
\(\chi_{4320}(2611,\cdot)\)
\(\chi_{4320}(3331,\cdot)\)
\(\chi_{4320}(3691,\cdot)\)
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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 ]
\((2431,3781,2081,3457)\) → \((-1,e\left(\frac{1}{8}\right),e\left(\frac{1}{3}\right),1)\)
| \(a\) |
\(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 4320 }(91, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(-1\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{5}{12}\right)\) |
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sage:chi.jacobi_sum(n)
