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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([13,125]))
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pari:[g,chi] = znchar(Mod(779,8450))
\(\chi_{8450}(129,\cdot)\)
\(\chi_{8450}(259,\cdot)\)
\(\chi_{8450}(389,\cdot)\)
\(\chi_{8450}(519,\cdot)\)
\(\chi_{8450}(779,\cdot)\)
\(\chi_{8450}(909,\cdot)\)
\(\chi_{8450}(1039,\cdot)\)
\(\chi_{8450}(1169,\cdot)\)
\(\chi_{8450}(1429,\cdot)\)
\(\chi_{8450}(1559,\cdot)\)
\(\chi_{8450}(1819,\cdot)\)
\(\chi_{8450}(2079,\cdot)\)
\(\chi_{8450}(2209,\cdot)\)
\(\chi_{8450}(2339,\cdot)\)
\(\chi_{8450}(2469,\cdot)\)
\(\chi_{8450}(2729,\cdot)\)
\(\chi_{8450}(2859,\cdot)\)
\(\chi_{8450}(2989,\cdot)\)
\(\chi_{8450}(3119,\cdot)\)
\(\chi_{8450}(3509,\cdot)\)
\(\chi_{8450}(3639,\cdot)\)
\(\chi_{8450}(3769,\cdot)\)
\(\chi_{8450}(4029,\cdot)\)
\(\chi_{8450}(4159,\cdot)\)
\(\chi_{8450}(4289,\cdot)\)
\(\chi_{8450}(4419,\cdot)\)
\(\chi_{8450}(4679,\cdot)\)
\(\chi_{8450}(4809,\cdot)\)
\(\chi_{8450}(4939,\cdot)\)
\(\chi_{8450}(5329,\cdot)\)
...
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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 ]
\((677,3551)\) → \((e\left(\frac{1}{10}\right),e\left(\frac{25}{26}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 8450 }(779, a) \) |
\(1\) | \(1\) | \(e\left(\frac{121}{130}\right)\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{56}{65}\right)\) | \(e\left(\frac{83}{130}\right)\) | \(e\left(\frac{89}{130}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{41}{130}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{103}{130}\right)\) | \(e\left(\frac{43}{65}\right)\) |
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sage:chi.jacobi_sum(n)
