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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6897, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([0,51,11]))
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pari:[g,chi] = znchar(Mod(1528,6897))
\(\chi_{6897}(274,\cdot)\)
\(\chi_{6897}(373,\cdot)\)
\(\chi_{6897}(901,\cdot)\)
\(\chi_{6897}(1000,\cdot)\)
\(\chi_{6897}(1528,\cdot)\)
\(\chi_{6897}(1627,\cdot)\)
\(\chi_{6897}(2155,\cdot)\)
\(\chi_{6897}(2254,\cdot)\)
\(\chi_{6897}(2881,\cdot)\)
\(\chi_{6897}(3409,\cdot)\)
\(\chi_{6897}(4036,\cdot)\)
\(\chi_{6897}(4135,\cdot)\)
\(\chi_{6897}(4663,\cdot)\)
\(\chi_{6897}(4762,\cdot)\)
\(\chi_{6897}(5290,\cdot)\)
\(\chi_{6897}(5389,\cdot)\)
\(\chi_{6897}(5917,\cdot)\)
\(\chi_{6897}(6016,\cdot)\)
\(\chi_{6897}(6544,\cdot)\)
\(\chi_{6897}(6643,\cdot)\)
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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 ]
\((2300,970,3631)\) → \((1,e\left(\frac{17}{22}\right),e\left(\frac{1}{6}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(13\) | \(14\) | \(16\) | \(17\) |
| \( \chi_{ 6897 }(1528, a) \) |
\(1\) | \(1\) | \(e\left(\frac{31}{33}\right)\) | \(e\left(\frac{29}{33}\right)\) | \(e\left(\frac{28}{33}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{26}{33}\right)\) | \(e\left(\frac{29}{33}\right)\) | \(e\left(\frac{23}{66}\right)\) | \(e\left(\frac{25}{33}\right)\) | \(e\left(\frac{35}{66}\right)\) |
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sage:chi.jacobi_sum(n)
