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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([50,57,30]))
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pari:[g,chi] = znchar(Mod(113,8550))
\(\chi_{8550}(113,\cdot)\)
\(\chi_{8550}(227,\cdot)\)
\(\chi_{8550}(797,\cdot)\)
\(\chi_{8550}(1253,\cdot)\)
\(\chi_{8550}(1823,\cdot)\)
\(\chi_{8550}(1937,\cdot)\)
\(\chi_{8550}(2963,\cdot)\)
\(\chi_{8550}(3533,\cdot)\)
\(\chi_{8550}(3647,\cdot)\)
\(\chi_{8550}(4217,\cdot)\)
\(\chi_{8550}(4673,\cdot)\)
\(\chi_{8550}(5927,\cdot)\)
\(\chi_{8550}(6383,\cdot)\)
\(\chi_{8550}(6953,\cdot)\)
\(\chi_{8550}(7067,\cdot)\)
\(\chi_{8550}(7637,\cdot)\)
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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 ]
\((1901,1027,1351)\) → \((e\left(\frac{5}{6}\right),e\left(\frac{19}{20}\right),-1)\)
| \(a\) |
\(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 8550 }(113, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{37}{60}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{7}{12}\right)\) |
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sage:chi.jacobi_sum(n)
