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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5850, base_ring=CyclotomicField(30))
M = H._module
chi = DirichletCharacter(H, M([20,21,5]))
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pari:[g,chi] = znchar(Mod(5659,5850))
\(\chi_{5850}(979,\cdot)\)
\(\chi_{5850}(1219,\cdot)\)
\(\chi_{5850}(2389,\cdot)\)
\(\chi_{5850}(3319,\cdot)\)
\(\chi_{5850}(3559,\cdot)\)
\(\chi_{5850}(4489,\cdot)\)
\(\chi_{5850}(4729,\cdot)\)
\(\chi_{5850}(5659,\cdot)\)
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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 ]
\((3251,3277,2251)\) → \((e\left(\frac{2}{3}\right),e\left(\frac{7}{10}\right),e\left(\frac{1}{6}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(7\) | \(11\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 5850 }(5659, a) \) |
\(1\) | \(1\) | \(1\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(-1\) |
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sage:chi.jacobi_sum(n)
