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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9075, base_ring=CyclotomicField(110))
M = H._module
chi = DirichletCharacter(H, M([0,44,86]))
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pari:[g,chi] = znchar(Mod(31,9075))
\(\chi_{9075}(31,\cdot)\)
\(\chi_{9075}(91,\cdot)\)
\(\chi_{9075}(136,\cdot)\)
\(\chi_{9075}(346,\cdot)\)
\(\chi_{9075}(916,\cdot)\)
\(\chi_{9075}(961,\cdot)\)
\(\chi_{9075}(1171,\cdot)\)
\(\chi_{9075}(1681,\cdot)\)
\(\chi_{9075}(1741,\cdot)\)
\(\chi_{9075}(1786,\cdot)\)
\(\chi_{9075}(1996,\cdot)\)
\(\chi_{9075}(2506,\cdot)\)
\(\chi_{9075}(2566,\cdot)\)
\(\chi_{9075}(2611,\cdot)\)
\(\chi_{9075}(2821,\cdot)\)
\(\chi_{9075}(3331,\cdot)\)
\(\chi_{9075}(3436,\cdot)\)
\(\chi_{9075}(3646,\cdot)\)
\(\chi_{9075}(4156,\cdot)\)
\(\chi_{9075}(4216,\cdot)\)
\(\chi_{9075}(4261,\cdot)\)
\(\chi_{9075}(4471,\cdot)\)
\(\chi_{9075}(4981,\cdot)\)
\(\chi_{9075}(5041,\cdot)\)
\(\chi_{9075}(5086,\cdot)\)
\(\chi_{9075}(5296,\cdot)\)
\(\chi_{9075}(5806,\cdot)\)
\(\chi_{9075}(5866,\cdot)\)
\(\chi_{9075}(5911,\cdot)\)
\(\chi_{9075}(6121,\cdot)\)
...
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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 ]
\((3026,727,5326)\) → \((1,e\left(\frac{2}{5}\right),e\left(\frac{43}{55}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) | \(23\) |
| \( \chi_{ 9075 }(31, a) \) |
\(1\) | \(1\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{26}{55}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{31}{55}\right)\) | \(e\left(\frac{36}{55}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{28}{55}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{7}{55}\right)\) |
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sage:chi.jacobi_sum(n)
