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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8450, base_ring=CyclotomicField(780))
M = H._module
chi = DirichletCharacter(H, M([156,695]))
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pari:[g,chi] = znchar(Mod(791,8450))
\(\chi_{8450}(11,\cdot)\)
\(\chi_{8450}(41,\cdot)\)
\(\chi_{8450}(71,\cdot)\)
\(\chi_{8450}(111,\cdot)\)
\(\chi_{8450}(141,\cdot)\)
\(\chi_{8450}(171,\cdot)\)
\(\chi_{8450}(241,\cdot)\)
\(\chi_{8450}(271,\cdot)\)
\(\chi_{8450}(331,\cdot)\)
\(\chi_{8450}(371,\cdot)\)
\(\chi_{8450}(431,\cdot)\)
\(\chi_{8450}(461,\cdot)\)
\(\chi_{8450}(531,\cdot)\)
\(\chi_{8450}(561,\cdot)\)
\(\chi_{8450}(591,\cdot)\)
\(\chi_{8450}(631,\cdot)\)
\(\chi_{8450}(661,\cdot)\)
\(\chi_{8450}(691,\cdot)\)
\(\chi_{8450}(721,\cdot)\)
\(\chi_{8450}(761,\cdot)\)
\(\chi_{8450}(791,\cdot)\)
\(\chi_{8450}(821,\cdot)\)
\(\chi_{8450}(891,\cdot)\)
\(\chi_{8450}(921,\cdot)\)
\(\chi_{8450}(981,\cdot)\)
\(\chi_{8450}(1021,\cdot)\)
\(\chi_{8450}(1081,\cdot)\)
\(\chi_{8450}(1111,\cdot)\)
\(\chi_{8450}(1181,\cdot)\)
\(\chi_{8450}(1211,\cdot)\)
...
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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 ]
\((677,3551)\) → \((e\left(\frac{1}{5}\right),e\left(\frac{139}{156}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 8450 }(791, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{173}{195}\right)\) | \(e\left(\frac{53}{156}\right)\) | \(e\left(\frac{151}{195}\right)\) | \(e\left(\frac{761}{780}\right)\) | \(e\left(\frac{269}{390}\right)\) | \(e\left(\frac{31}{60}\right)\) | \(e\left(\frac{59}{260}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{43}{65}\right)\) | \(e\left(\frac{8}{195}\right)\) |
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sage:chi.jacobi_sum(n)
