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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7742, base_ring=CyclotomicField(546))
M = H._module
chi = DirichletCharacter(H, M([416,98]))
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pari:[g,chi] = znchar(Mod(625,7742))
\(\chi_{7742}(9,\cdot)\)
\(\chi_{7742}(25,\cdot)\)
\(\chi_{7742}(81,\cdot)\)
\(\chi_{7742}(95,\cdot)\)
\(\chi_{7742}(123,\cdot)\)
\(\chi_{7742}(163,\cdot)\)
\(\chi_{7742}(207,\cdot)\)
\(\chi_{7742}(431,\cdot)\)
\(\chi_{7742}(485,\cdot)\)
\(\chi_{7742}(487,\cdot)\)
\(\chi_{7742}(625,\cdot)\)
\(\chi_{7742}(683,\cdot)\)
\(\chi_{7742}(737,\cdot)\)
\(\chi_{7742}(751,\cdot)\)
\(\chi_{7742}(821,\cdot)\)
\(\chi_{7742}(919,\cdot)\)
\(\chi_{7742}(1031,\cdot)\)
\(\chi_{7742}(1103,\cdot)\)
\(\chi_{7742}(1115,\cdot)\)
\(\chi_{7742}(1131,\cdot)\)
\(\chi_{7742}(1187,\cdot)\)
\(\chi_{7742}(1201,\cdot)\)
\(\chi_{7742}(1229,\cdot)\)
\(\chi_{7742}(1269,\cdot)\)
\(\chi_{7742}(1283,\cdot)\)
\(\chi_{7742}(1313,\cdot)\)
\(\chi_{7742}(1467,\cdot)\)
\(\chi_{7742}(1591,\cdot)\)
\(\chi_{7742}(1593,\cdot)\)
\(\chi_{7742}(1731,\cdot)\)
...
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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 ]
\((2845,4901)\) → \((e\left(\frac{16}{21}\right),e\left(\frac{7}{39}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) |
| \( \chi_{ 7742 }(625, a) \) |
\(1\) | \(1\) | \(e\left(\frac{257}{273}\right)\) | \(e\left(\frac{61}{273}\right)\) | \(e\left(\frac{241}{273}\right)\) | \(e\left(\frac{62}{91}\right)\) | \(e\left(\frac{67}{273}\right)\) | \(e\left(\frac{15}{91}\right)\) | \(e\left(\frac{223}{273}\right)\) | \(e\left(\frac{16}{39}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{122}{273}\right)\) |
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sage:chi.jacobi_sum(n)
