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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8732, base_ring=CyclotomicField(174))
M = H._module
chi = DirichletCharacter(H, M([0,29,102]))
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pari:[g,chi] = znchar(Mod(4393,8732))
\(\chi_{8732}(85,\cdot)\)
\(\chi_{8732}(381,\cdot)\)
\(\chi_{8732}(529,\cdot)\)
\(\chi_{8732}(677,\cdot)\)
\(\chi_{8732}(841,\cdot)\)
\(\chi_{8732}(973,\cdot)\)
\(\chi_{8732}(989,\cdot)\)
\(\chi_{8732}(1137,\cdot)\)
\(\chi_{8732}(1285,\cdot)\)
\(\chi_{8732}(1433,\cdot)\)
\(\chi_{8732}(1877,\cdot)\)
\(\chi_{8732}(2009,\cdot)\)
\(\chi_{8732}(2025,\cdot)\)
\(\chi_{8732}(2173,\cdot)\)
\(\chi_{8732}(2305,\cdot)\)
\(\chi_{8732}(2321,\cdot)\)
\(\chi_{8732}(2601,\cdot)\)
\(\chi_{8732}(2617,\cdot)\)
\(\chi_{8732}(2749,\cdot)\)
\(\chi_{8732}(2765,\cdot)\)
\(\chi_{8732}(2913,\cdot)\)
\(\chi_{8732}(3045,\cdot)\)
\(\chi_{8732}(3193,\cdot)\)
\(\chi_{8732}(3357,\cdot)\)
\(\chi_{8732}(3785,\cdot)\)
\(\chi_{8732}(3801,\cdot)\)
\(\chi_{8732}(4097,\cdot)\)
\(\chi_{8732}(4393,\cdot)\)
\(\chi_{8732}(4525,\cdot)\)
\(\chi_{8732}(4541,\cdot)\)
...
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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 ]
\((4367,1889,297)\) → \((1,e\left(\frac{1}{6}\right),e\left(\frac{17}{29}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 8732 }(4393, a) \) |
\(1\) | \(1\) | \(e\left(\frac{56}{87}\right)\) | \(e\left(\frac{61}{174}\right)\) | \(e\left(\frac{77}{87}\right)\) | \(e\left(\frac{25}{87}\right)\) | \(e\left(\frac{19}{29}\right)\) | \(e\left(\frac{37}{174}\right)\) | \(e\left(\frac{173}{174}\right)\) | \(e\left(\frac{107}{174}\right)\) | \(e\left(\frac{19}{174}\right)\) | \(e\left(\frac{46}{87}\right)\) |
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sage:chi.jacobi_sum(n)
