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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2864, base_ring=CyclotomicField(178))
M = H._module
chi = DirichletCharacter(H, M([89,0,132]))
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pari:[g,chi] = znchar(Mod(607,2864))
\(\chi_{2864}(15,\cdot)\)
\(\chi_{2864}(31,\cdot)\)
\(\chi_{2864}(47,\cdot)\)
\(\chi_{2864}(95,\cdot)\)
\(\chi_{2864}(191,\cdot)\)
\(\chi_{2864}(239,\cdot)\)
\(\chi_{2864}(255,\cdot)\)
\(\chi_{2864}(287,\cdot)\)
\(\chi_{2864}(303,\cdot)\)
\(\chi_{2864}(335,\cdot)\)
\(\chi_{2864}(351,\cdot)\)
\(\chi_{2864}(367,\cdot)\)
\(\chi_{2864}(383,\cdot)\)
\(\chi_{2864}(415,\cdot)\)
\(\chi_{2864}(447,\cdot)\)
\(\chi_{2864}(479,\cdot)\)
\(\chi_{2864}(511,\cdot)\)
\(\chi_{2864}(527,\cdot)\)
\(\chi_{2864}(559,\cdot)\)
\(\chi_{2864}(607,\cdot)\)
\(\chi_{2864}(719,\cdot)\)
\(\chi_{2864}(735,\cdot)\)
\(\chi_{2864}(767,\cdot)\)
\(\chi_{2864}(783,\cdot)\)
\(\chi_{2864}(799,\cdot)\)
\(\chi_{2864}(863,\cdot)\)
\(\chi_{2864}(911,\cdot)\)
\(\chi_{2864}(943,\cdot)\)
\(\chi_{2864}(959,\cdot)\)
\(\chi_{2864}(975,\cdot)\)
...
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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 ]
\((1791,2149,897)\) → \((-1,1,e\left(\frac{66}{89}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 2864 }(607, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{105}{178}\right)\) | \(e\left(\frac{30}{89}\right)\) | \(e\left(\frac{55}{178}\right)\) | \(e\left(\frac{16}{89}\right)\) | \(e\left(\frac{111}{178}\right)\) | \(e\left(\frac{48}{89}\right)\) | \(e\left(\frac{165}{178}\right)\) | \(e\left(\frac{9}{89}\right)\) | \(e\left(\frac{97}{178}\right)\) | \(e\left(\frac{80}{89}\right)\) |
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sage:chi.jacobi_sum(n)
