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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4864, base_ring=CyclotomicField(288))
M = H._module
chi = DirichletCharacter(H, M([144,189,128]))
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pari:[g,chi] = znchar(Mod(1415,4864))
\(\chi_{4864}(23,\cdot)\)
\(\chi_{4864}(55,\cdot)\)
\(\chi_{4864}(119,\cdot)\)
\(\chi_{4864}(199,\cdot)\)
\(\chi_{4864}(215,\cdot)\)
\(\chi_{4864}(263,\cdot)\)
\(\chi_{4864}(327,\cdot)\)
\(\chi_{4864}(359,\cdot)\)
\(\chi_{4864}(423,\cdot)\)
\(\chi_{4864}(503,\cdot)\)
\(\chi_{4864}(519,\cdot)\)
\(\chi_{4864}(567,\cdot)\)
\(\chi_{4864}(631,\cdot)\)
\(\chi_{4864}(663,\cdot)\)
\(\chi_{4864}(727,\cdot)\)
\(\chi_{4864}(807,\cdot)\)
\(\chi_{4864}(823,\cdot)\)
\(\chi_{4864}(871,\cdot)\)
\(\chi_{4864}(935,\cdot)\)
\(\chi_{4864}(967,\cdot)\)
\(\chi_{4864}(1031,\cdot)\)
\(\chi_{4864}(1111,\cdot)\)
\(\chi_{4864}(1127,\cdot)\)
\(\chi_{4864}(1175,\cdot)\)
\(\chi_{4864}(1239,\cdot)\)
\(\chi_{4864}(1271,\cdot)\)
\(\chi_{4864}(1335,\cdot)\)
\(\chi_{4864}(1415,\cdot)\)
\(\chi_{4864}(1431,\cdot)\)
\(\chi_{4864}(1479,\cdot)\)
...
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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 ]
\((3839,2053,4353)\) → \((-1,e\left(\frac{21}{32}\right),e\left(\frac{4}{9}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(21\) | \(23\) |
| \( \chi_{ 4864 }(1415, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{71}{288}\right)\) | \(e\left(\frac{221}{288}\right)\) | \(e\left(\frac{35}{48}\right)\) | \(e\left(\frac{71}{144}\right)\) | \(e\left(\frac{59}{96}\right)\) | \(e\left(\frac{19}{288}\right)\) | \(e\left(\frac{1}{72}\right)\) | \(e\left(\frac{59}{72}\right)\) | \(e\left(\frac{281}{288}\right)\) | \(e\left(\frac{83}{144}\right)\) |
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sage:chi.jacobi_sum(n)
