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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(16830, base_ring=CyclotomicField(240))
M = H._module
chi = DirichletCharacter(H, M([160,60,144,45]))
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pari:[g,chi] = znchar(Mod(8017,16830))
\(\chi_{16830}(367,\cdot)\)
\(\chi_{16830}(643,\cdot)\)
\(\chi_{16830}(907,\cdot)\)
\(\chi_{16830}(1093,\cdot)\)
\(\chi_{16830}(1417,\cdot)\)
\(\chi_{16830}(1897,\cdot)\)
\(\chi_{16830}(2407,\cdot)\)
\(\chi_{16830}(2623,\cdot)\)
\(\chi_{16830}(3067,\cdot)\)
\(\chi_{16830}(3193,\cdot)\)
\(\chi_{16830}(3463,\cdot)\)
\(\chi_{16830}(4057,\cdot)\)
\(\chi_{16830}(4723,\cdot)\)
\(\chi_{16830}(5107,\cdot)\)
\(\chi_{16830}(5173,\cdot)\)
\(\chi_{16830}(5443,\cdot)\)
\(\chi_{16830}(5503,\cdot)\)
\(\chi_{16830}(6097,\cdot)\)
\(\chi_{16830}(6253,\cdot)\)
\(\chi_{16830}(6637,\cdot)\)
\(\chi_{16830}(6703,\cdot)\)
\(\chi_{16830}(6763,\cdot)\)
\(\chi_{16830}(7027,\cdot)\)
\(\chi_{16830}(7033,\cdot)\)
\(\chi_{16830}(7483,\cdot)\)
\(\chi_{16830}(7627,\cdot)\)
\(\chi_{16830}(8017,\cdot)\)
\(\chi_{16830}(8167,\cdot)\)
\(\chi_{16830}(8233,\cdot)\)
\(\chi_{16830}(8563,\cdot)\)
...
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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 ]
\((7481,3367,1531,8911)\) → \((e\left(\frac{2}{3}\right),i,e\left(\frac{3}{5}\right),e\left(\frac{3}{16}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(7\) | \(13\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) | \(47\) |
| \( \chi_{ 16830 }(8017, a) \) |
\(1\) | \(1\) | \(e\left(\frac{43}{240}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{43}{48}\right)\) | \(e\left(\frac{193}{240}\right)\) | \(e\left(\frac{149}{240}\right)\) | \(e\left(\frac{51}{80}\right)\) | \(e\left(\frac{47}{240}\right)\) | \(e\left(\frac{19}{24}\right)\) | \(e\left(\frac{7}{15}\right)\) |
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sage:chi.jacobi_sum(n)
