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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([513,62]))
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pari:[g,chi] = znchar(Mod(421,9386))
\(\chi_{9386}(21,\cdot)\)
\(\chi_{9386}(109,\cdot)\)
\(\chi_{9386}(135,\cdot)\)
\(\chi_{9386}(203,\cdot)\)
\(\chi_{9386}(281,\cdot)\)
\(\chi_{9386}(317,\cdot)\)
\(\chi_{9386}(395,\cdot)\)
\(\chi_{9386}(421,\cdot)\)
\(\chi_{9386}(447,\cdot)\)
\(\chi_{9386}(489,\cdot)\)
\(\chi_{9386}(515,\cdot)\)
\(\chi_{9386}(603,\cdot)\)
\(\chi_{9386}(629,\cdot)\)
\(\chi_{9386}(697,\cdot)\)
\(\chi_{9386}(775,\cdot)\)
\(\chi_{9386}(801,\cdot)\)
\(\chi_{9386}(811,\cdot)\)
\(\chi_{9386}(827,\cdot)\)
\(\chi_{9386}(889,\cdot)\)
\(\chi_{9386}(915,\cdot)\)
\(\chi_{9386}(941,\cdot)\)
\(\chi_{9386}(983,\cdot)\)
\(\chi_{9386}(1009,\cdot)\)
\(\chi_{9386}(1097,\cdot)\)
\(\chi_{9386}(1123,\cdot)\)
\(\chi_{9386}(1191,\cdot)\)
\(\chi_{9386}(1269,\cdot)\)
\(\chi_{9386}(1295,\cdot)\)
\(\chi_{9386}(1305,\cdot)\)
\(\chi_{9386}(1321,\cdot)\)
...
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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 ]
\((1445,3251)\) → \((-i,e\left(\frac{31}{342}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(15\) | \(17\) | \(21\) | \(23\) | \(25\) |
| \( \chi_{ 9386 }(421, a) \) |
\(1\) | \(1\) | \(e\left(\frac{205}{342}\right)\) | \(e\left(\frac{533}{684}\right)\) | \(e\left(\frac{193}{228}\right)\) | \(e\left(\frac{34}{171}\right)\) | \(e\left(\frac{113}{228}\right)\) | \(e\left(\frac{259}{684}\right)\) | \(e\left(\frac{337}{342}\right)\) | \(e\left(\frac{305}{684}\right)\) | \(e\left(\frac{251}{342}\right)\) | \(e\left(\frac{191}{342}\right)\) |
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sage:chi.jacobi_sum(n)
