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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9386, base_ring=CyclotomicField(684))
M = H._module
chi = DirichletCharacter(H, M([285,158]))
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pari:[g,chi] = znchar(Mod(71,9386))
\(\chi_{9386}(15,\cdot)\)
\(\chi_{9386}(33,\cdot)\)
\(\chi_{9386}(59,\cdot)\)
\(\chi_{9386}(67,\cdot)\)
\(\chi_{9386}(71,\cdot)\)
\(\chi_{9386}(89,\cdot)\)
\(\chi_{9386}(97,\cdot)\)
\(\chi_{9386}(167,\cdot)\)
\(\chi_{9386}(219,\cdot)\)
\(\chi_{9386}(345,\cdot)\)
\(\chi_{9386}(383,\cdot)\)
\(\chi_{9386}(431,\cdot)\)
\(\chi_{9386}(509,\cdot)\)
\(\chi_{9386}(527,\cdot)\)
\(\chi_{9386}(553,\cdot)\)
\(\chi_{9386}(561,\cdot)\)
\(\chi_{9386}(565,\cdot)\)
\(\chi_{9386}(583,\cdot)\)
\(\chi_{9386}(591,\cdot)\)
\(\chi_{9386}(661,\cdot)\)
\(\chi_{9386}(713,\cdot)\)
\(\chi_{9386}(839,\cdot)\)
\(\chi_{9386}(877,\cdot)\)
\(\chi_{9386}(925,\cdot)\)
\(\chi_{9386}(1003,\cdot)\)
\(\chi_{9386}(1047,\cdot)\)
\(\chi_{9386}(1059,\cdot)\)
\(\chi_{9386}(1077,\cdot)\)
\(\chi_{9386}(1085,\cdot)\)
\(\chi_{9386}(1155,\cdot)\)
...
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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 ]
\((1445,3251)\) → \((e\left(\frac{5}{12}\right),e\left(\frac{79}{342}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(15\) | \(17\) | \(21\) | \(23\) | \(25\) |
| \( \chi_{ 9386 }(71, a) \) |
\(1\) | \(1\) | \(e\left(\frac{265}{342}\right)\) | \(e\left(\frac{233}{684}\right)\) | \(e\left(\frac{53}{228}\right)\) | \(e\left(\frac{94}{171}\right)\) | \(e\left(\frac{109}{228}\right)\) | \(e\left(\frac{79}{684}\right)\) | \(e\left(\frac{13}{342}\right)\) | \(e\left(\frac{5}{684}\right)\) | \(e\left(\frac{305}{342}\right)\) | \(e\left(\frac{233}{342}\right)\) |
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sage:chi.jacobi_sum(n)
