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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5850, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([10,42,5]))
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pari:[g,chi] = znchar(Mod(2459,5850))
\(\chi_{5850}(59,\cdot)\)
\(\chi_{5850}(119,\cdot)\)
\(\chi_{5850}(479,\cdot)\)
\(\chi_{5850}(1229,\cdot)\)
\(\chi_{5850}(1289,\cdot)\)
\(\chi_{5850}(1319,\cdot)\)
\(\chi_{5850}(2459,\cdot)\)
\(\chi_{5850}(2489,\cdot)\)
\(\chi_{5850}(2819,\cdot)\)
\(\chi_{5850}(3569,\cdot)\)
\(\chi_{5850}(3629,\cdot)\)
\(\chi_{5850}(3659,\cdot)\)
\(\chi_{5850}(3989,\cdot)\)
\(\chi_{5850}(4739,\cdot)\)
\(\chi_{5850}(4829,\cdot)\)
\(\chi_{5850}(5159,\cdot)\)
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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 ]
\((3251,3277,2251)\) → \((e\left(\frac{1}{6}\right),e\left(\frac{7}{10}\right),e\left(\frac{1}{12}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(7\) | \(11\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 5850 }(2459, a) \) |
\(1\) | \(1\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{1}{60}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{41}{60}\right)\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{43}{60}\right)\) | \(e\left(\frac{1}{3}\right)\) |
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sage:chi.jacobi_sum(n)
