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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([156,15]))
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pari:[g,chi] = znchar(Mod(2371,8450))
\(\chi_{8450}(21,\cdot)\)
\(\chi_{8450}(31,\cdot)\)
\(\chi_{8450}(161,\cdot)\)
\(\chi_{8450}(281,\cdot)\)
\(\chi_{8450}(291,\cdot)\)
\(\chi_{8450}(411,\cdot)\)
\(\chi_{8450}(421,\cdot)\)
\(\chi_{8450}(541,\cdot)\)
\(\chi_{8450}(671,\cdot)\)
\(\chi_{8450}(681,\cdot)\)
\(\chi_{8450}(811,\cdot)\)
\(\chi_{8450}(931,\cdot)\)
\(\chi_{8450}(941,\cdot)\)
\(\chi_{8450}(1061,\cdot)\)
\(\chi_{8450}(1071,\cdot)\)
\(\chi_{8450}(1191,\cdot)\)
\(\chi_{8450}(1321,\cdot)\)
\(\chi_{8450}(1331,\cdot)\)
\(\chi_{8450}(1461,\cdot)\)
\(\chi_{8450}(1581,\cdot)\)
\(\chi_{8450}(1711,\cdot)\)
\(\chi_{8450}(1721,\cdot)\)
\(\chi_{8450}(1841,\cdot)\)
\(\chi_{8450}(1971,\cdot)\)
\(\chi_{8450}(1981,\cdot)\)
\(\chi_{8450}(2111,\cdot)\)
\(\chi_{8450}(2231,\cdot)\)
\(\chi_{8450}(2241,\cdot)\)
\(\chi_{8450}(2361,\cdot)\)
\(\chi_{8450}(2371,\cdot)\)
...
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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 ]
\((677,3551)\) → \((e\left(\frac{3}{5}\right),e\left(\frac{3}{52}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 8450 }(2371, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{23}{65}\right)\) | \(e\left(\frac{9}{52}\right)\) | \(e\left(\frac{46}{65}\right)\) | \(e\left(\frac{141}{260}\right)\) | \(e\left(\frac{29}{130}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{137}{260}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{4}{65}\right)\) | \(e\left(\frac{33}{65}\right)\) |
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sage:chi.jacobi_sum(n)
