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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(33957, base_ring=CyclotomicField(210))
M = H._module
chi = DirichletCharacter(H, M([140,25,168]))
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pari:[g,chi] = znchar(Mod(9133,33957))
\(\chi_{33957}(31,\cdot)\)
\(\chi_{33957}(313,\cdot)\)
\(\chi_{33957}(2236,\cdot)\)
\(\chi_{33957}(2677,\cdot)\)
\(\chi_{33957}(3118,\cdot)\)
\(\chi_{33957}(4282,\cdot)\)
\(\chi_{33957}(4723,\cdot)\)
\(\chi_{33957}(4882,\cdot)\)
\(\chi_{33957}(6928,\cdot)\)
\(\chi_{33957}(7087,\cdot)\)
\(\chi_{33957}(7969,\cdot)\)
\(\chi_{33957}(9133,\cdot)\)
\(\chi_{33957}(9574,\cdot)\)
\(\chi_{33957}(9733,\cdot)\)
\(\chi_{33957}(10015,\cdot)\)
\(\chi_{33957}(11779,\cdot)\)
\(\chi_{33957}(11938,\cdot)\)
\(\chi_{33957}(12379,\cdot)\)
\(\chi_{33957}(12820,\cdot)\)
\(\chi_{33957}(13984,\cdot)\)
\(\chi_{33957}(14584,\cdot)\)
\(\chi_{33957}(14866,\cdot)\)
\(\chi_{33957}(16630,\cdot)\)
\(\chi_{33957}(17230,\cdot)\)
\(\chi_{33957}(17671,\cdot)\)
\(\chi_{33957}(18835,\cdot)\)
\(\chi_{33957}(19276,\cdot)\)
\(\chi_{33957}(19435,\cdot)\)
\(\chi_{33957}(19717,\cdot)\)
\(\chi_{33957}(21481,\cdot)\)
...
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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 ]
\((18866,14752,24697)\) → \((e\left(\frac{2}{3}\right),e\left(\frac{5}{42}\right),e\left(\frac{4}{5}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(13\) | \(16\) | \(17\) | \(19\) | \(20\) |
| \( \chi_{ 33957 }(9133, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{59}{105}\right)\) | \(e\left(\frac{13}{105}\right)\) | \(e\left(\frac{69}{70}\right)\) | \(e\left(\frac{24}{35}\right)\) | \(e\left(\frac{23}{42}\right)\) | \(e\left(\frac{13}{210}\right)\) | \(e\left(\frac{26}{105}\right)\) | \(e\left(\frac{37}{210}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{23}{210}\right)\) |
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sage:chi.jacobi_sum(n)
