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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8954, base_ring=CyclotomicField(660))
M = H._module
chi = DirichletCharacter(H, M([318,275]))
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pari:[g,chi] = znchar(Mod(393,8954))
\(\chi_{8954}(29,\cdot)\)
\(\chi_{8954}(51,\cdot)\)
\(\chi_{8954}(171,\cdot)\)
\(\chi_{8954}(193,\cdot)\)
\(\chi_{8954}(325,\cdot)\)
\(\chi_{8954}(347,\cdot)\)
\(\chi_{8954}(393,\cdot)\)
\(\chi_{8954}(415,\cdot)\)
\(\chi_{8954}(541,\cdot)\)
\(\chi_{8954}(547,\cdot)\)
\(\chi_{8954}(563,\cdot)\)
\(\chi_{8954}(569,\cdot)\)
\(\chi_{8954}(689,\cdot)\)
\(\chi_{8954}(695,\cdot)\)
\(\chi_{8954}(711,\cdot)\)
\(\chi_{8954}(843,\cdot)\)
\(\chi_{8954}(865,\cdot)\)
\(\chi_{8954}(985,\cdot)\)
\(\chi_{8954}(1007,\cdot)\)
\(\chi_{8954}(1139,\cdot)\)
\(\chi_{8954}(1161,\cdot)\)
\(\chi_{8954}(1229,\cdot)\)
\(\chi_{8954}(1355,\cdot)\)
\(\chi_{8954}(1361,\cdot)\)
\(\chi_{8954}(1377,\cdot)\)
\(\chi_{8954}(1383,\cdot)\)
\(\chi_{8954}(1503,\cdot)\)
\(\chi_{8954}(1509,\cdot)\)
\(\chi_{8954}(1525,\cdot)\)
\(\chi_{8954}(1531,\cdot)\)
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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 ]
\((1333,3147)\) → \((e\left(\frac{53}{110}\right),e\left(\frac{5}{12}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) | \(23\) |
| \( \chi_{ 8954 }(393, a) \) |
\(1\) | \(1\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{157}{660}\right)\) | \(e\left(\frac{233}{330}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{163}{660}\right)\) | \(e\left(\frac{311}{660}\right)\) | \(e\left(\frac{347}{660}\right)\) | \(e\left(\frac{379}{660}\right)\) | \(e\left(\frac{31}{33}\right)\) | \(e\left(\frac{43}{44}\right)\) |
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sage:chi.jacobi_sum(n)
