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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(25410, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([0,0,22,6]))
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pari:[g,chi] = znchar(Mod(18811,25410))
\(\chi_{25410}(331,\cdot)\)
\(\chi_{25410}(991,\cdot)\)
\(\chi_{25410}(2641,\cdot)\)
\(\chi_{25410}(3301,\cdot)\)
\(\chi_{25410}(4951,\cdot)\)
\(\chi_{25410}(5611,\cdot)\)
\(\chi_{25410}(7921,\cdot)\)
\(\chi_{25410}(9571,\cdot)\)
\(\chi_{25410}(10231,\cdot)\)
\(\chi_{25410}(11881,\cdot)\)
\(\chi_{25410}(12541,\cdot)\)
\(\chi_{25410}(14191,\cdot)\)
\(\chi_{25410}(14851,\cdot)\)
\(\chi_{25410}(16501,\cdot)\)
\(\chi_{25410}(17161,\cdot)\)
\(\chi_{25410}(18811,\cdot)\)
\(\chi_{25410}(19471,\cdot)\)
\(\chi_{25410}(21121,\cdot)\)
\(\chi_{25410}(23431,\cdot)\)
\(\chi_{25410}(24091,\cdot)\)
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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 ]
\((8471,15247,14521,7141)\) → \((1,1,e\left(\frac{1}{3}\right),e\left(\frac{1}{11}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) | \(47\) |
| \( \chi_{ 25410 }(18811, a) \) |
\(1\) | \(1\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{26}{33}\right)\) | \(e\left(\frac{7}{33}\right)\) | \(e\left(\frac{1}{33}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{5}{33}\right)\) | \(e\left(\frac{16}{33}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{19}{33}\right)\) |
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sage:chi.jacobi_sum(n)
