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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8450, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([54,25]))
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pari:[g,chi] = znchar(Mod(19,8450))
\(\chi_{8450}(19,\cdot)\)
\(\chi_{8450}(89,\cdot)\)
\(\chi_{8450}(319,\cdot)\)
\(\chi_{8450}(1709,\cdot)\)
\(\chi_{8450}(1779,\cdot)\)
\(\chi_{8450}(1939,\cdot)\)
\(\chi_{8450}(2009,\cdot)\)
\(\chi_{8450}(3469,\cdot)\)
\(\chi_{8450}(3629,\cdot)\)
\(\chi_{8450}(5089,\cdot)\)
\(\chi_{8450}(5159,\cdot)\)
\(\chi_{8450}(5319,\cdot)\)
\(\chi_{8450}(5389,\cdot)\)
\(\chi_{8450}(6779,\cdot)\)
\(\chi_{8450}(7009,\cdot)\)
\(\chi_{8450}(7079,\cdot)\)
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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 ]
\((677,3551)\) → \((e\left(\frac{9}{10}\right),e\left(\frac{5}{12}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 8450 }(19, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{19}{60}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{17}{60}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{7}{15}\right)\) |
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sage:chi.jacobi_sum(n)
