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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9800, base_ring=CyclotomicField(84))
M = H._module
chi = DirichletCharacter(H, M([0,42,63,50]))
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pari:[g,chi] = znchar(Mod(6093,9800))
\(\chi_{9800}(157,\cdot)\)
\(\chi_{9800}(493,\cdot)\)
\(\chi_{9800}(957,\cdot)\)
\(\chi_{9800}(1557,\cdot)\)
\(\chi_{9800}(2357,\cdot)\)
\(\chi_{9800}(2693,\cdot)\)
\(\chi_{9800}(2957,\cdot)\)
\(\chi_{9800}(3293,\cdot)\)
\(\chi_{9800}(3757,\cdot)\)
\(\chi_{9800}(4093,\cdot)\)
\(\chi_{9800}(4357,\cdot)\)
\(\chi_{9800}(4693,\cdot)\)
\(\chi_{9800}(5157,\cdot)\)
\(\chi_{9800}(5493,\cdot)\)
\(\chi_{9800}(5757,\cdot)\)
\(\chi_{9800}(6093,\cdot)\)
\(\chi_{9800}(6557,\cdot)\)
\(\chi_{9800}(6893,\cdot)\)
\(\chi_{9800}(7157,\cdot)\)
\(\chi_{9800}(7493,\cdot)\)
\(\chi_{9800}(8293,\cdot)\)
\(\chi_{9800}(8893,\cdot)\)
\(\chi_{9800}(9357,\cdot)\)
\(\chi_{9800}(9693,\cdot)\)
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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 ]
\((7351,4901,1177,5001)\) → \((1,-1,-i,e\left(\frac{25}{42}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(27\) | \(29\) | \(31\) |
| \( \chi_{ 9800 }(6093, a) \) |
\(1\) | \(1\) | \(e\left(\frac{29}{84}\right)\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{13}{42}\right)\) | \(e\left(\frac{11}{28}\right)\) | \(e\left(\frac{53}{84}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{73}{84}\right)\) | \(e\left(\frac{1}{28}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{1}{6}\right)\) |
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sage:chi.jacobi_sum(n)
