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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9900, base_ring=CyclotomicField(30))
M = H._module
chi = DirichletCharacter(H, M([0,5,18,27]))
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pari:[g,chi] = znchar(Mod(6221,9900))
\(\chi_{9900}(41,\cdot)\)
\(\chi_{9900}(2081,\cdot)\)
\(\chi_{9900}(2921,\cdot)\)
\(\chi_{9900}(3341,\cdot)\)
\(\chi_{9900}(3461,\cdot)\)
\(\chi_{9900}(6221,\cdot)\)
\(\chi_{9900}(6761,\cdot)\)
\(\chi_{9900}(8681,\cdot)\)
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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 ]
\((4951,5501,2377,4501)\) → \((1,e\left(\frac{1}{6}\right),e\left(\frac{3}{5}\right),e\left(\frac{9}{10}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(7\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 9900 }(6221, a) \) |
\(1\) | \(1\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(-1\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{1}{6}\right)\) |
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sage:chi.jacobi_sum(n)
