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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([3,10]))
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pari:[g,chi] = znchar(Mod(3527,8450))
\(\chi_{8450}(23,\cdot)\)
\(\chi_{8450}(147,\cdot)\)
\(\chi_{8450}(823,\cdot)\)
\(\chi_{8450}(1037,\cdot)\)
\(\chi_{8450}(1713,\cdot)\)
\(\chi_{8450}(1837,\cdot)\)
\(\chi_{8450}(2513,\cdot)\)
\(\chi_{8450}(2727,\cdot)\)
\(\chi_{8450}(3403,\cdot)\)
\(\chi_{8450}(3527,\cdot)\)
\(\chi_{8450}(4203,\cdot)\)
\(\chi_{8450}(4417,\cdot)\)
\(\chi_{8450}(5217,\cdot)\)
\(\chi_{8450}(6783,\cdot)\)
\(\chi_{8450}(7583,\cdot)\)
\(\chi_{8450}(7797,\cdot)\)
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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 ]
\((677,3551)\) → \((e\left(\frac{1}{20}\right),e\left(\frac{1}{6}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 8450 }(3527, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{1}{60}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{59}{60}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{23}{30}\right)\) |
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sage:chi.jacobi_sum(n)
