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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(12138, base_ring=CyclotomicField(102))
M = H._module
chi = DirichletCharacter(H, M([0,68,87]))
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pari:[g,chi] = znchar(Mod(6493,12138))
\(\chi_{12138}(67,\cdot)\)
\(\chi_{12138}(373,\cdot)\)
\(\chi_{12138}(781,\cdot)\)
\(\chi_{12138}(1087,\cdot)\)
\(\chi_{12138}(1495,\cdot)\)
\(\chi_{12138}(1801,\cdot)\)
\(\chi_{12138}(2209,\cdot)\)
\(\chi_{12138}(2515,\cdot)\)
\(\chi_{12138}(2923,\cdot)\)
\(\chi_{12138}(3229,\cdot)\)
\(\chi_{12138}(3637,\cdot)\)
\(\chi_{12138}(3943,\cdot)\)
\(\chi_{12138}(4351,\cdot)\)
\(\chi_{12138}(4657,\cdot)\)
\(\chi_{12138}(5065,\cdot)\)
\(\chi_{12138}(5371,\cdot)\)
\(\chi_{12138}(6085,\cdot)\)
\(\chi_{12138}(6493,\cdot)\)
\(\chi_{12138}(6799,\cdot)\)
\(\chi_{12138}(7207,\cdot)\)
\(\chi_{12138}(7921,\cdot)\)
\(\chi_{12138}(8227,\cdot)\)
\(\chi_{12138}(8635,\cdot)\)
\(\chi_{12138}(8941,\cdot)\)
\(\chi_{12138}(9349,\cdot)\)
\(\chi_{12138}(9655,\cdot)\)
\(\chi_{12138}(10063,\cdot)\)
\(\chi_{12138}(10369,\cdot)\)
\(\chi_{12138}(10777,\cdot)\)
\(\chi_{12138}(11083,\cdot)\)
...
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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 ]
\((8093,10405,9829)\) → \((1,e\left(\frac{2}{3}\right),e\left(\frac{29}{34}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 12138 }(6493, a) \) |
\(1\) | \(1\) | \(e\left(\frac{67}{102}\right)\) | \(e\left(\frac{29}{102}\right)\) | \(e\left(\frac{3}{17}\right)\) | \(e\left(\frac{14}{51}\right)\) | \(e\left(\frac{55}{102}\right)\) | \(e\left(\frac{16}{51}\right)\) | \(e\left(\frac{21}{34}\right)\) | \(e\left(\frac{35}{102}\right)\) | \(e\left(\frac{37}{102}\right)\) | \(e\left(\frac{29}{34}\right)\) |
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sage:chi.jacobi_sum(n)
