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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(16830, base_ring=CyclotomicField(120))
M = H._module
chi = DirichletCharacter(H, M([80,60,24,105]))
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pari:[g,chi] = znchar(Mod(5119,16830))
\(\chi_{16830}(49,\cdot)\)
\(\chi_{16830}(229,\cdot)\)
\(\chi_{16830}(1039,\cdot)\)
\(\chi_{16830}(1879,\cdot)\)
\(\chi_{16830}(2599,\cdot)\)
\(\chi_{16830}(3589,\cdot)\)
\(\chi_{16830}(3919,\cdot)\)
\(\chi_{16830}(4129,\cdot)\)
\(\chi_{16830}(5119,\cdot)\)
\(\chi_{16830}(5449,\cdot)\)
\(\chi_{16830}(5659,\cdot)\)
\(\chi_{16830}(5839,\cdot)\)
\(\chi_{16830}(6169,\cdot)\)
\(\chi_{16830}(6649,\cdot)\)
\(\chi_{16830}(6979,\cdot)\)
\(\chi_{16830}(7159,\cdot)\)
\(\chi_{16830}(7879,\cdot)\)
\(\chi_{16830}(9409,\cdot)\)
\(\chi_{16830}(9529,\cdot)\)
\(\chi_{16830}(10939,\cdot)\)
\(\chi_{16830}(11059,\cdot)\)
\(\chi_{16830}(11779,\cdot)\)
\(\chi_{16830}(12589,\cdot)\)
\(\chi_{16830}(12769,\cdot)\)
\(\chi_{16830}(13099,\cdot)\)
\(\chi_{16830}(13489,\cdot)\)
\(\chi_{16830}(13819,\cdot)\)
\(\chi_{16830}(14809,\cdot)\)
\(\chi_{16830}(15019,\cdot)\)
\(\chi_{16830}(15349,\cdot)\)
...
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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 ]
\((7481,3367,1531,8911)\) → \((e\left(\frac{2}{3}\right),-1,e\left(\frac{1}{5}\right),e\left(\frac{7}{8}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(7\) | \(13\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) | \(47\) |
| \( \chi_{ 16830 }(5119, a) \) |
\(1\) | \(1\) | \(e\left(\frac{23}{120}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{23}{24}\right)\) | \(e\left(\frac{53}{120}\right)\) | \(e\left(\frac{49}{120}\right)\) | \(e\left(\frac{31}{40}\right)\) | \(e\left(\frac{67}{120}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{4}{15}\right)\) |
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sage:chi.jacobi_sum(n)
