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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([52,25]))
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pari:[g,chi] = znchar(Mod(831,8450))
\(\chi_{8450}(181,\cdot)\)
\(\chi_{8450}(311,\cdot)\)
\(\chi_{8450}(441,\cdot)\)
\(\chi_{8450}(571,\cdot)\)
\(\chi_{8450}(831,\cdot)\)
\(\chi_{8450}(961,\cdot)\)
\(\chi_{8450}(1091,\cdot)\)
\(\chi_{8450}(1221,\cdot)\)
\(\chi_{8450}(1481,\cdot)\)
\(\chi_{8450}(1611,\cdot)\)
\(\chi_{8450}(1741,\cdot)\)
\(\chi_{8450}(1871,\cdot)\)
\(\chi_{8450}(2131,\cdot)\)
\(\chi_{8450}(2261,\cdot)\)
\(\chi_{8450}(2391,\cdot)\)
\(\chi_{8450}(2521,\cdot)\)
\(\chi_{8450}(2781,\cdot)\)
\(\chi_{8450}(2911,\cdot)\)
\(\chi_{8450}(3171,\cdot)\)
\(\chi_{8450}(3431,\cdot)\)
\(\chi_{8450}(3561,\cdot)\)
\(\chi_{8450}(3691,\cdot)\)
\(\chi_{8450}(3821,\cdot)\)
\(\chi_{8450}(4081,\cdot)\)
\(\chi_{8450}(4211,\cdot)\)
\(\chi_{8450}(4341,\cdot)\)
\(\chi_{8450}(4471,\cdot)\)
\(\chi_{8450}(4861,\cdot)\)
\(\chi_{8450}(4991,\cdot)\)
\(\chi_{8450}(5121,\cdot)\)
...
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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 ]
\((677,3551)\) → \((e\left(\frac{2}{5}\right),e\left(\frac{5}{26}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 8450 }(831, a) \) |
\(1\) | \(1\) | \(e\left(\frac{42}{65}\right)\) | \(e\left(\frac{15}{26}\right)\) | \(e\left(\frac{19}{65}\right)\) | \(e\left(\frac{27}{130}\right)\) | \(e\left(\frac{18}{65}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{29}{130}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{61}{65}\right)\) | \(e\left(\frac{32}{65}\right)\) |
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sage:chi.jacobi_sum(n)
