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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8041, base_ring=CyclotomicField(336))
M = H._module
chi = DirichletCharacter(H, M([0,273,104]))
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pari:[g,chi] = znchar(Mod(12,8041))
\(\chi_{8041}(12,\cdot)\)
\(\chi_{8041}(177,\cdot)\)
\(\chi_{8041}(243,\cdot)\)
\(\chi_{8041}(320,\cdot)\)
\(\chi_{8041}(364,\cdot)\)
\(\chi_{8041}(507,\cdot)\)
\(\chi_{8041}(760,\cdot)\)
\(\chi_{8041}(793,\cdot)\)
\(\chi_{8041}(804,\cdot)\)
\(\chi_{8041}(958,\cdot)\)
\(\chi_{8041}(980,\cdot)\)
\(\chi_{8041}(1431,\cdot)\)
\(\chi_{8041}(1508,\cdot)\)
\(\chi_{8041}(1574,\cdot)\)
\(\chi_{8041}(1695,\cdot)\)
\(\chi_{8041}(1706,\cdot)\)
\(\chi_{8041}(1739,\cdot)\)
\(\chi_{8041}(1882,\cdot)\)
\(\chi_{8041}(1926,\cdot)\)
\(\chi_{8041}(2047,\cdot)\)
\(\chi_{8041}(2069,\cdot)\)
\(\chi_{8041}(2135,\cdot)\)
\(\chi_{8041}(2179,\cdot)\)
\(\chi_{8041}(2256,\cdot)\)
\(\chi_{8041}(2377,\cdot)\)
\(\chi_{8041}(2454,\cdot)\)
\(\chi_{8041}(2608,\cdot)\)
\(\chi_{8041}(2641,\cdot)\)
\(\chi_{8041}(2696,\cdot)\)
\(\chi_{8041}(2828,\cdot)\)
...
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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 ]
\((6580,2366,562)\) → \((1,e\left(\frac{13}{16}\right),e\left(\frac{13}{42}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(12\) |
| \( \chi_{ 8041 }(12, a) \) |
\(1\) | \(1\) | \(e\left(\frac{41}{56}\right)\) | \(e\left(\frac{41}{336}\right)\) | \(e\left(\frac{13}{28}\right)\) | \(e\left(\frac{269}{336}\right)\) | \(e\left(\frac{41}{48}\right)\) | \(e\left(\frac{37}{48}\right)\) | \(e\left(\frac{11}{56}\right)\) | \(e\left(\frac{41}{168}\right)\) | \(e\left(\frac{179}{336}\right)\) | \(e\left(\frac{197}{336}\right)\) |
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sage:chi.jacobi_sum(n)
