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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4830, base_ring=CyclotomicField(132))
M = H._module
chi = DirichletCharacter(H, M([66,99,22,126]))
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pari:[g,chi] = znchar(Mod(773,4830))
\(\chi_{4830}(17,\cdot)\)
\(\chi_{4830}(143,\cdot)\)
\(\chi_{4830}(227,\cdot)\)
\(\chi_{4830}(383,\cdot)\)
\(\chi_{4830}(467,\cdot)\)
\(\chi_{4830}(563,\cdot)\)
\(\chi_{4830}(677,\cdot)\)
\(\chi_{4830}(773,\cdot)\)
\(\chi_{4830}(803,\cdot)\)
\(\chi_{4830}(983,\cdot)\)
\(\chi_{4830}(1193,\cdot)\)
\(\chi_{4830}(1307,\cdot)\)
\(\chi_{4830}(1433,\cdot)\)
\(\chi_{4830}(1487,\cdot)\)
\(\chi_{4830}(1643,\cdot)\)
\(\chi_{4830}(1907,\cdot)\)
\(\chi_{4830}(1937,\cdot)\)
\(\chi_{4830}(2273,\cdot)\)
\(\chi_{4830}(2357,\cdot)\)
\(\chi_{4830}(2453,\cdot)\)
\(\chi_{4830}(2537,\cdot)\)
\(\chi_{4830}(2567,\cdot)\)
\(\chi_{4830}(2747,\cdot)\)
\(\chi_{4830}(2777,\cdot)\)
\(\chi_{4830}(2873,\cdot)\)
\(\chi_{4830}(2903,\cdot)\)
\(\chi_{4830}(2987,\cdot)\)
\(\chi_{4830}(3323,\cdot)\)
\(\chi_{4830}(3377,\cdot)\)
\(\chi_{4830}(3503,\cdot)\)
...
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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 ]
\((3221,967,2761,1891)\) → \((-1,-i,e\left(\frac{1}{6}\right),e\left(\frac{21}{22}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(11\) | \(13\) | \(17\) | \(19\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) | \(47\) |
| \( \chi_{ 4830 }(773, a) \) |
\(1\) | \(1\) | \(e\left(\frac{25}{33}\right)\) | \(e\left(\frac{5}{44}\right)\) | \(e\left(\frac{13}{132}\right)\) | \(e\left(\frac{43}{66}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{59}{66}\right)\) | \(e\left(\frac{17}{132}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{1}{44}\right)\) | \(e\left(\frac{1}{12}\right)\) |
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sage:chi.jacobi_sum(n)
