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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8550, base_ring=CyclotomicField(180))
M = H._module
chi = DirichletCharacter(H, M([30,27,80]))
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pari:[g,chi] = znchar(Mod(2783,8550))
\(\chi_{8550}(47,\cdot)\)
\(\chi_{8550}(137,\cdot)\)
\(\chi_{8550}(347,\cdot)\)
\(\chi_{8550}(473,\cdot)\)
\(\chi_{8550}(833,\cdot)\)
\(\chi_{8550}(1073,\cdot)\)
\(\chi_{8550}(1127,\cdot)\)
\(\chi_{8550}(1163,\cdot)\)
\(\chi_{8550}(1373,\cdot)\)
\(\chi_{8550}(1517,\cdot)\)
\(\chi_{8550}(1847,\cdot)\)
\(\chi_{8550}(2153,\cdot)\)
\(\chi_{8550}(2183,\cdot)\)
\(\chi_{8550}(2783,\cdot)\)
\(\chi_{8550}(2837,\cdot)\)
\(\chi_{8550}(2867,\cdot)\)
\(\chi_{8550}(2873,\cdot)\)
\(\chi_{8550}(3083,\cdot)\)
\(\chi_{8550}(3227,\cdot)\)
\(\chi_{8550}(3467,\cdot)\)
\(\chi_{8550}(3767,\cdot)\)
\(\chi_{8550}(3863,\cdot)\)
\(\chi_{8550}(4253,\cdot)\)
\(\chi_{8550}(4547,\cdot)\)
\(\chi_{8550}(4577,\cdot)\)
\(\chi_{8550}(4583,\cdot)\)
\(\chi_{8550}(4937,\cdot)\)
\(\chi_{8550}(5177,\cdot)\)
\(\chi_{8550}(5267,\cdot)\)
\(\chi_{8550}(5477,\cdot)\)
...
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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 ]
\((1901,1027,1351)\) → \((e\left(\frac{1}{6}\right),e\left(\frac{3}{20}\right),e\left(\frac{4}{9}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 8550 }(2783, a) \) |
\(1\) | \(1\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{73}{180}\right)\) | \(e\left(\frac{161}{180}\right)\) | \(e\left(\frac{67}{180}\right)\) | \(e\left(\frac{1}{45}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{19}{90}\right)\) | \(e\left(\frac{1}{36}\right)\) |
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sage:chi.jacobi_sum(n)
