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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8550, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([50,27,0]))
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pari:[g,chi] = znchar(Mod(1787,8550))
\(\chi_{8550}(77,\cdot)\)
\(\chi_{8550}(533,\cdot)\)
\(\chi_{8550}(1103,\cdot)\)
\(\chi_{8550}(1217,\cdot)\)
\(\chi_{8550}(1787,\cdot)\)
\(\chi_{8550}(2813,\cdot)\)
\(\chi_{8550}(2927,\cdot)\)
\(\chi_{8550}(3497,\cdot)\)
\(\chi_{8550}(3953,\cdot)\)
\(\chi_{8550}(4523,\cdot)\)
\(\chi_{8550}(4637,\cdot)\)
\(\chi_{8550}(5663,\cdot)\)
\(\chi_{8550}(6233,\cdot)\)
\(\chi_{8550}(6347,\cdot)\)
\(\chi_{8550}(6917,\cdot)\)
\(\chi_{8550}(7373,\cdot)\)
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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 ]
\((1901,1027,1351)\) → \((e\left(\frac{5}{6}\right),e\left(\frac{9}{20}\right),1)\)
| \(a\) |
\(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 8550 }(1787, a) \) |
\(1\) | \(1\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{7}{60}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{1}{12}\right)\) |
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sage:chi.jacobi_sum(n)
