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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7260, base_ring=CyclotomicField(220))
M = H._module
chi = DirichletCharacter(H, M([0,110,55,98]))
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pari:[g,chi] = znchar(Mod(17,7260))
\(\chi_{7260}(17,\cdot)\)
\(\chi_{7260}(173,\cdot)\)
\(\chi_{7260}(293,\cdot)\)
\(\chi_{7260}(413,\cdot)\)
\(\chi_{7260}(437,\cdot)\)
\(\chi_{7260}(497,\cdot)\)
\(\chi_{7260}(557,\cdot)\)
\(\chi_{7260}(677,\cdot)\)
\(\chi_{7260}(833,\cdot)\)
\(\chi_{7260}(893,\cdot)\)
\(\chi_{7260}(953,\cdot)\)
\(\chi_{7260}(1073,\cdot)\)
\(\chi_{7260}(1097,\cdot)\)
\(\chi_{7260}(1157,\cdot)\)
\(\chi_{7260}(1217,\cdot)\)
\(\chi_{7260}(1337,\cdot)\)
\(\chi_{7260}(1493,\cdot)\)
\(\chi_{7260}(1553,\cdot)\)
\(\chi_{7260}(1733,\cdot)\)
\(\chi_{7260}(1757,\cdot)\)
\(\chi_{7260}(1817,\cdot)\)
\(\chi_{7260}(1877,\cdot)\)
\(\chi_{7260}(1997,\cdot)\)
\(\chi_{7260}(2153,\cdot)\)
\(\chi_{7260}(2213,\cdot)\)
\(\chi_{7260}(2273,\cdot)\)
\(\chi_{7260}(2477,\cdot)\)
\(\chi_{7260}(2537,\cdot)\)
\(\chi_{7260}(2657,\cdot)\)
\(\chi_{7260}(2813,\cdot)\)
...
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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 ]
\((3631,4841,4357,7141)\) → \((1,-1,i,e\left(\frac{49}{110}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(7\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 7260 }(17, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{81}{220}\right)\) | \(e\left(\frac{163}{220}\right)\) | \(e\left(\frac{127}{220}\right)\) | \(e\left(\frac{26}{55}\right)\) | \(e\left(\frac{19}{44}\right)\) | \(e\left(\frac{63}{110}\right)\) | \(e\left(\frac{17}{55}\right)\) | \(e\left(\frac{211}{220}\right)\) | \(e\left(\frac{41}{55}\right)\) | \(e\left(\frac{39}{44}\right)\) |
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sage:chi.jacobi_sum(n)
