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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8550, base_ring=CyclotomicField(90))
M = H._module
chi = DirichletCharacter(H, M([0,27,25]))
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pari:[g,chi] = znchar(Mod(3889,8550))
\(\chi_{8550}(109,\cdot)\)
\(\chi_{8550}(469,\cdot)\)
\(\chi_{8550}(1009,\cdot)\)
\(\chi_{8550}(1459,\cdot)\)
\(\chi_{8550}(1819,\cdot)\)
\(\chi_{8550}(2179,\cdot)\)
\(\chi_{8550}(2359,\cdot)\)
\(\chi_{8550}(2719,\cdot)\)
\(\chi_{8550}(3169,\cdot)\)
\(\chi_{8550}(3259,\cdot)\)
\(\chi_{8550}(3529,\cdot)\)
\(\chi_{8550}(3889,\cdot)\)
\(\chi_{8550}(4069,\cdot)\)
\(\chi_{8550}(4429,\cdot)\)
\(\chi_{8550}(4879,\cdot)\)
\(\chi_{8550}(4969,\cdot)\)
\(\chi_{8550}(5239,\cdot)\)
\(\chi_{8550}(5779,\cdot)\)
\(\chi_{8550}(6139,\cdot)\)
\(\chi_{8550}(6589,\cdot)\)
\(\chi_{8550}(6679,\cdot)\)
\(\chi_{8550}(7309,\cdot)\)
\(\chi_{8550}(7489,\cdot)\)
\(\chi_{8550}(8389,\cdot)\)
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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 ]
\((1901,1027,1351)\) → \((1,e\left(\frac{3}{10}\right),e\left(\frac{5}{18}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 8550 }(3889, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{4}{45}\right)\) | \(e\left(\frac{61}{90}\right)\) | \(e\left(\frac{77}{90}\right)\) | \(e\left(\frac{29}{90}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{73}{90}\right)\) | \(e\left(\frac{17}{18}\right)\) |
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sage:chi.jacobi_sum(n)
