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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8450, base_ring=CyclotomicField(390))
M = H._module
chi = DirichletCharacter(H, M([78,205]))
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pari:[g,chi] = znchar(Mod(491,8450))
\(\chi_{8450}(121,\cdot)\)
\(\chi_{8450}(231,\cdot)\)
\(\chi_{8450}(381,\cdot)\)
\(\chi_{8450}(491,\cdot)\)
\(\chi_{8450}(511,\cdot)\)
\(\chi_{8450}(621,\cdot)\)
\(\chi_{8450}(641,\cdot)\)
\(\chi_{8450}(771,\cdot)\)
\(\chi_{8450}(881,\cdot)\)
\(\chi_{8450}(1011,\cdot)\)
\(\chi_{8450}(1031,\cdot)\)
\(\chi_{8450}(1141,\cdot)\)
\(\chi_{8450}(1271,\cdot)\)
\(\chi_{8450}(1291,\cdot)\)
\(\chi_{8450}(1421,\cdot)\)
\(\chi_{8450}(1531,\cdot)\)
\(\chi_{8450}(1661,\cdot)\)
\(\chi_{8450}(1681,\cdot)\)
\(\chi_{8450}(1791,\cdot)\)
\(\chi_{8450}(1811,\cdot)\)
\(\chi_{8450}(1921,\cdot)\)
\(\chi_{8450}(1941,\cdot)\)
\(\chi_{8450}(2071,\cdot)\)
\(\chi_{8450}(2181,\cdot)\)
\(\chi_{8450}(2311,\cdot)\)
\(\chi_{8450}(2331,\cdot)\)
\(\chi_{8450}(2441,\cdot)\)
\(\chi_{8450}(2461,\cdot)\)
\(\chi_{8450}(2571,\cdot)\)
\(\chi_{8450}(2591,\cdot)\)
...
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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 ]
\((677,3551)\) → \((e\left(\frac{1}{5}\right),e\left(\frac{41}{78}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 8450 }(491, a) \) |
\(1\) | \(1\) | \(e\left(\frac{113}{195}\right)\) | \(e\left(\frac{19}{78}\right)\) | \(e\left(\frac{31}{195}\right)\) | \(e\left(\frac{133}{390}\right)\) | \(e\left(\frac{67}{195}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{107}{130}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{48}{65}\right)\) | \(e\left(\frac{83}{195}\right)\) |
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sage:chi.jacobi_sum(n)
