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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(25410, base_ring=CyclotomicField(30))
M = H._module
chi = DirichletCharacter(H, M([0,15,20,12]))
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pari:[g,chi] = znchar(Mod(24469,25410))
\(\chi_{25410}(3469,\cdot)\)
\(\chi_{25410}(5569,\cdot)\)
\(\chi_{25410}(8479,\cdot)\)
\(\chi_{25410}(9949,\cdot)\)
\(\chi_{25410}(14359,\cdot)\)
\(\chi_{25410}(16459,\cdot)\)
\(\chi_{25410}(22999,\cdot)\)
\(\chi_{25410}(24469,\cdot)\)
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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 ]
\((8471,15247,14521,7141)\) → \((1,-1,e\left(\frac{2}{3}\right),e\left(\frac{2}{5}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) | \(47\) |
| \( \chi_{ 25410 }(24469, a) \) |
\(1\) | \(1\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(-1\) | \(e\left(\frac{1}{30}\right)\) |
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sage:chi.jacobi_sum(n)
