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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9386, base_ring=CyclotomicField(114))
M = H._module
chi = DirichletCharacter(H, M([57,10]))
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pari:[g,chi] = znchar(Mod(6889,9386))
\(\chi_{9386}(311,\cdot)\)
\(\chi_{9386}(467,\cdot)\)
\(\chi_{9386}(805,\cdot)\)
\(\chi_{9386}(961,\cdot)\)
\(\chi_{9386}(1299,\cdot)\)
\(\chi_{9386}(1455,\cdot)\)
\(\chi_{9386}(1793,\cdot)\)
\(\chi_{9386}(1949,\cdot)\)
\(\chi_{9386}(2287,\cdot)\)
\(\chi_{9386}(2443,\cdot)\)
\(\chi_{9386}(2781,\cdot)\)
\(\chi_{9386}(2937,\cdot)\)
\(\chi_{9386}(3275,\cdot)\)
\(\chi_{9386}(3431,\cdot)\)
\(\chi_{9386}(3769,\cdot)\)
\(\chi_{9386}(3925,\cdot)\)
\(\chi_{9386}(4419,\cdot)\)
\(\chi_{9386}(4757,\cdot)\)
\(\chi_{9386}(4913,\cdot)\)
\(\chi_{9386}(5251,\cdot)\)
\(\chi_{9386}(5407,\cdot)\)
\(\chi_{9386}(5745,\cdot)\)
\(\chi_{9386}(5901,\cdot)\)
\(\chi_{9386}(6239,\cdot)\)
\(\chi_{9386}(6395,\cdot)\)
\(\chi_{9386}(6733,\cdot)\)
\(\chi_{9386}(6889,\cdot)\)
\(\chi_{9386}(7227,\cdot)\)
\(\chi_{9386}(7383,\cdot)\)
\(\chi_{9386}(7721,\cdot)\)
...
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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 ]
\((1445,3251)\) → \((-1,e\left(\frac{5}{57}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(15\) | \(17\) | \(21\) | \(23\) | \(25\) |
| \( \chi_{ 9386 }(6889, a) \) |
\(1\) | \(1\) | \(e\left(\frac{11}{57}\right)\) | \(e\left(\frac{97}{114}\right)\) | \(e\left(\frac{25}{38}\right)\) | \(e\left(\frac{22}{57}\right)\) | \(e\left(\frac{17}{38}\right)\) | \(e\left(\frac{5}{114}\right)\) | \(e\left(\frac{47}{57}\right)\) | \(e\left(\frac{97}{114}\right)\) | \(e\left(\frac{46}{57}\right)\) | \(e\left(\frac{40}{57}\right)\) |
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sage:chi.jacobi_sum(n)
