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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2057, base_ring=CyclotomicField(110))
M = H._module
chi = DirichletCharacter(H, M([1,0]))
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pari:[g,chi] = znchar(Mod(970,2057))
\(\chi_{2057}(18,\cdot)\)
\(\chi_{2057}(35,\cdot)\)
\(\chi_{2057}(52,\cdot)\)
\(\chi_{2057}(171,\cdot)\)
\(\chi_{2057}(205,\cdot)\)
\(\chi_{2057}(222,\cdot)\)
\(\chi_{2057}(358,\cdot)\)
\(\chi_{2057}(392,\cdot)\)
\(\chi_{2057}(409,\cdot)\)
\(\chi_{2057}(426,\cdot)\)
\(\chi_{2057}(545,\cdot)\)
\(\chi_{2057}(579,\cdot)\)
\(\chi_{2057}(613,\cdot)\)
\(\chi_{2057}(732,\cdot)\)
\(\chi_{2057}(783,\cdot)\)
\(\chi_{2057}(800,\cdot)\)
\(\chi_{2057}(919,\cdot)\)
\(\chi_{2057}(953,\cdot)\)
\(\chi_{2057}(970,\cdot)\)
\(\chi_{2057}(987,\cdot)\)
\(\chi_{2057}(1106,\cdot)\)
\(\chi_{2057}(1140,\cdot)\)
\(\chi_{2057}(1157,\cdot)\)
\(\chi_{2057}(1174,\cdot)\)
\(\chi_{2057}(1293,\cdot)\)
\(\chi_{2057}(1327,\cdot)\)
\(\chi_{2057}(1344,\cdot)\)
\(\chi_{2057}(1361,\cdot)\)
\(\chi_{2057}(1480,\cdot)\)
\(\chi_{2057}(1514,\cdot)\)
...
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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 ]
\((970,122)\) → \((e\left(\frac{1}{110}\right),1)\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(12\) |
| \( \chi_{ 2057 }(970, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{1}{110}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{1}{55}\right)\) | \(e\left(\frac{37}{55}\right)\) | \(e\left(\frac{89}{110}\right)\) | \(e\left(\frac{7}{110}\right)\) | \(e\left(\frac{3}{110}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{9}{11}\right)\) |
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sage:chi.jacobi_sum(n)
