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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2565, base_ring=CyclotomicField(18))
M = H._module
chi = DirichletCharacter(H, M([17,0,4]))
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pari:[g,chi] = znchar(Mod(2201,2565))
\(\chi_{2565}(131,\cdot)\)
\(\chi_{2565}(821,\cdot)\)
\(\chi_{2565}(1031,\cdot)\)
\(\chi_{2565}(1586,\cdot)\)
\(\chi_{2565}(1811,\cdot)\)
\(\chi_{2565}(2201,\cdot)\)
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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 ]
\((191,1027,1351)\) → \((e\left(\frac{17}{18}\right),1,e\left(\frac{2}{9}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(22\) |
| \( \chi_{ 2565 }(2201, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(-1\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{1}{9}\right)\) |
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sage:chi.jacobi_sum(n)
