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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(12138, base_ring=CyclotomicField(204))
M = H._module
chi = DirichletCharacter(H, M([0,68,57]))
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pari:[g,chi] = znchar(Mod(667,12138))
\(\chi_{12138}(319,\cdot)\)
\(\chi_{12138}(361,\cdot)\)
\(\chi_{12138}(625,\cdot)\)
\(\chi_{12138}(667,\cdot)\)
\(\chi_{12138}(1033,\cdot)\)
\(\chi_{12138}(1075,\cdot)\)
\(\chi_{12138}(1339,\cdot)\)
\(\chi_{12138}(1381,\cdot)\)
\(\chi_{12138}(1747,\cdot)\)
\(\chi_{12138}(1789,\cdot)\)
\(\chi_{12138}(2053,\cdot)\)
\(\chi_{12138}(2095,\cdot)\)
\(\chi_{12138}(2461,\cdot)\)
\(\chi_{12138}(2503,\cdot)\)
\(\chi_{12138}(2767,\cdot)\)
\(\chi_{12138}(2809,\cdot)\)
\(\chi_{12138}(3175,\cdot)\)
\(\chi_{12138}(3481,\cdot)\)
\(\chi_{12138}(3523,\cdot)\)
\(\chi_{12138}(3889,\cdot)\)
\(\chi_{12138}(3931,\cdot)\)
\(\chi_{12138}(4195,\cdot)\)
\(\chi_{12138}(4237,\cdot)\)
\(\chi_{12138}(4603,\cdot)\)
\(\chi_{12138}(4645,\cdot)\)
\(\chi_{12138}(4909,\cdot)\)
\(\chi_{12138}(5317,\cdot)\)
\(\chi_{12138}(5359,\cdot)\)
\(\chi_{12138}(5623,\cdot)\)
\(\chi_{12138}(5665,\cdot)\)
...
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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 ]
\((8093,10405,9829)\) → \((1,e\left(\frac{1}{3}\right),e\left(\frac{19}{68}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 12138 }(667, a) \) |
\(1\) | \(1\) | \(e\left(\frac{133}{204}\right)\) | \(e\left(\frac{155}{204}\right)\) | \(e\left(\frac{13}{17}\right)\) | \(e\left(\frac{59}{102}\right)\) | \(e\left(\frac{199}{204}\right)\) | \(e\left(\frac{31}{102}\right)\) | \(e\left(\frac{63}{68}\right)\) | \(e\left(\frac{173}{204}\right)\) | \(e\left(\frac{145}{204}\right)\) | \(e\left(\frac{53}{68}\right)\) |
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sage:chi.jacobi_sum(n)
