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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(254144, base_ring=CyclotomicField(2736))
M = H._module
chi = DirichletCharacter(H, M([1368,1539,0,136]))
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pari:[g,chi] = znchar(Mod(5083,254144))
\(\chi_{254144}(67,\cdot)\)
\(\chi_{254144}(155,\cdot)\)
\(\chi_{254144}(243,\cdot)\)
\(\chi_{254144}(507,\cdot)\)
\(\chi_{254144}(1123,\cdot)\)
\(\chi_{254144}(1211,\cdot)\)
\(\chi_{254144}(1739,\cdot)\)
\(\chi_{254144}(1827,\cdot)\)
\(\chi_{254144}(1915,\cdot)\)
\(\chi_{254144}(2179,\cdot)\)
\(\chi_{254144}(2795,\cdot)\)
\(\chi_{254144}(2883,\cdot)\)
\(\chi_{254144}(3411,\cdot)\)
\(\chi_{254144}(3499,\cdot)\)
\(\chi_{254144}(3587,\cdot)\)
\(\chi_{254144}(3851,\cdot)\)
\(\chi_{254144}(4467,\cdot)\)
\(\chi_{254144}(4555,\cdot)\)
\(\chi_{254144}(5083,\cdot)\)
\(\chi_{254144}(5171,\cdot)\)
\(\chi_{254144}(5259,\cdot)\)
\(\chi_{254144}(5523,\cdot)\)
\(\chi_{254144}(6139,\cdot)\)
\(\chi_{254144}(6227,\cdot)\)
\(\chi_{254144}(6755,\cdot)\)
\(\chi_{254144}(6843,\cdot)\)
\(\chi_{254144}(6931,\cdot)\)
\(\chi_{254144}(7195,\cdot)\)
\(\chi_{254144}(7811,\cdot)\)
\(\chi_{254144}(7899,\cdot)\)
...
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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 ]
\((166783,174725,69313,14081)\) → \((-1,e\left(\frac{9}{16}\right),1,e\left(\frac{17}{342}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(13\) | \(15\) | \(17\) | \(21\) | \(23\) | \(25\) |
| \( \chi_{ 254144 }(5083, a) \) |
\(1\) | \(1\) | \(e\left(\frac{265}{2736}\right)\) | \(e\left(\frac{259}{2736}\right)\) | \(e\left(\frac{265}{456}\right)\) | \(e\left(\frac{265}{1368}\right)\) | \(e\left(\frac{2165}{2736}\right)\) | \(e\left(\frac{131}{684}\right)\) | \(e\left(\frac{673}{684}\right)\) | \(e\left(\frac{1855}{2736}\right)\) | \(e\left(\frac{865}{1368}\right)\) | \(e\left(\frac{259}{1368}\right)\) |
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sage:chi.jacobi_sum(n)
