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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8450, base_ring=CyclotomicField(130))
M = H._module
chi = DirichletCharacter(H, M([39,60]))
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pari:[g,chi] = znchar(Mod(2939,8450))
\(\chi_{8450}(79,\cdot)\)
\(\chi_{8450}(209,\cdot)\)
\(\chi_{8450}(469,\cdot)\)
\(\chi_{8450}(729,\cdot)\)
\(\chi_{8450}(859,\cdot)\)
\(\chi_{8450}(989,\cdot)\)
\(\chi_{8450}(1119,\cdot)\)
\(\chi_{8450}(1379,\cdot)\)
\(\chi_{8450}(1509,\cdot)\)
\(\chi_{8450}(1639,\cdot)\)
\(\chi_{8450}(1769,\cdot)\)
\(\chi_{8450}(2159,\cdot)\)
\(\chi_{8450}(2289,\cdot)\)
\(\chi_{8450}(2419,\cdot)\)
\(\chi_{8450}(2679,\cdot)\)
\(\chi_{8450}(2809,\cdot)\)
\(\chi_{8450}(2939,\cdot)\)
\(\chi_{8450}(3069,\cdot)\)
\(\chi_{8450}(3329,\cdot)\)
\(\chi_{8450}(3459,\cdot)\)
\(\chi_{8450}(3589,\cdot)\)
\(\chi_{8450}(3979,\cdot)\)
\(\chi_{8450}(4109,\cdot)\)
\(\chi_{8450}(4239,\cdot)\)
\(\chi_{8450}(4369,\cdot)\)
\(\chi_{8450}(4629,\cdot)\)
\(\chi_{8450}(4759,\cdot)\)
\(\chi_{8450}(4889,\cdot)\)
\(\chi_{8450}(5019,\cdot)\)
\(\chi_{8450}(5279,\cdot)\)
...
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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 ]
\((677,3551)\) → \((e\left(\frac{3}{10}\right),e\left(\frac{6}{13}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 8450 }(2939, a) \) |
\(1\) | \(1\) | \(e\left(\frac{43}{130}\right)\) | \(e\left(\frac{23}{26}\right)\) | \(e\left(\frac{43}{65}\right)\) | \(e\left(\frac{22}{65}\right)\) | \(e\left(\frac{37}{130}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{14}{65}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{129}{130}\right)\) | \(e\left(\frac{4}{65}\right)\) |
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sage:chi.jacobi_sum(n)
