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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8450, base_ring=CyclotomicField(260))
M = H._module
chi = DirichletCharacter(H, M([13,175]))
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pari:[g,chi] = znchar(Mod(1227,8450))
\(\chi_{8450}(73,\cdot)\)
\(\chi_{8450}(187,\cdot)\)
\(\chi_{8450}(203,\cdot)\)
\(\chi_{8450}(317,\cdot)\)
\(\chi_{8450}(333,\cdot)\)
\(\chi_{8450}(447,\cdot)\)
\(\chi_{8450}(463,\cdot)\)
\(\chi_{8450}(723,\cdot)\)
\(\chi_{8450}(837,\cdot)\)
\(\chi_{8450}(853,\cdot)\)
\(\chi_{8450}(967,\cdot)\)
\(\chi_{8450}(983,\cdot)\)
\(\chi_{8450}(1097,\cdot)\)
\(\chi_{8450}(1227,\cdot)\)
\(\chi_{8450}(1373,\cdot)\)
\(\chi_{8450}(1487,\cdot)\)
\(\chi_{8450}(1503,\cdot)\)
\(\chi_{8450}(1617,\cdot)\)
\(\chi_{8450}(1633,\cdot)\)
\(\chi_{8450}(1747,\cdot)\)
\(\chi_{8450}(1763,\cdot)\)
\(\chi_{8450}(1877,\cdot)\)
\(\chi_{8450}(2023,\cdot)\)
\(\chi_{8450}(2137,\cdot)\)
\(\chi_{8450}(2153,\cdot)\)
\(\chi_{8450}(2283,\cdot)\)
\(\chi_{8450}(2397,\cdot)\)
\(\chi_{8450}(2413,\cdot)\)
\(\chi_{8450}(2527,\cdot)\)
\(\chi_{8450}(2673,\cdot)\)
...
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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 ]
\((677,3551)\) → \((e\left(\frac{1}{20}\right),e\left(\frac{35}{52}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 8450 }(1227, a) \) |
\(1\) | \(1\) | \(e\left(\frac{211}{260}\right)\) | \(e\left(\frac{7}{26}\right)\) | \(e\left(\frac{81}{130}\right)\) | \(e\left(\frac{33}{260}\right)\) | \(e\left(\frac{239}{260}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{21}{260}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{113}{260}\right)\) | \(e\left(\frac{3}{130}\right)\) |
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sage:chi.jacobi_sum(n)
