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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5175, base_ring=CyclotomicField(30))
M = H._module
chi = DirichletCharacter(H, M([10,9,0]))
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pari:[g,chi] = znchar(Mod(139,5175))
\(\chi_{5175}(139,\cdot)\)
\(\chi_{5175}(484,\cdot)\)
\(\chi_{5175}(1519,\cdot)\)
\(\chi_{5175}(2209,\cdot)\)
\(\chi_{5175}(2554,\cdot)\)
\(\chi_{5175}(3244,\cdot)\)
\(\chi_{5175}(3589,\cdot)\)
\(\chi_{5175}(4279,\cdot)\)
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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 ]
\((4601,3727,2926)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{3}{10}\right),1)\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 5175 }(139, a) \) |
\(1\) | \(1\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{2}{5}\right)\) |
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sage:chi.jacobi_sum(n)
