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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([0,0,165,202]))
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pari:[g,chi] = znchar(Mod(13,7260))
\(\chi_{7260}(13,\cdot)\)
\(\chi_{7260}(73,\cdot)\)
\(\chi_{7260}(193,\cdot)\)
\(\chi_{7260}(217,\cdot)\)
\(\chi_{7260}(277,\cdot)\)
\(\chi_{7260}(337,\cdot)\)
\(\chi_{7260}(613,\cdot)\)
\(\chi_{7260}(673,\cdot)\)
\(\chi_{7260}(733,\cdot)\)
\(\chi_{7260}(853,\cdot)\)
\(\chi_{7260}(877,\cdot)\)
\(\chi_{7260}(937,\cdot)\)
\(\chi_{7260}(997,\cdot)\)
\(\chi_{7260}(1117,\cdot)\)
\(\chi_{7260}(1273,\cdot)\)
\(\chi_{7260}(1333,\cdot)\)
\(\chi_{7260}(1393,\cdot)\)
\(\chi_{7260}(1513,\cdot)\)
\(\chi_{7260}(1537,\cdot)\)
\(\chi_{7260}(1597,\cdot)\)
\(\chi_{7260}(1657,\cdot)\)
\(\chi_{7260}(1777,\cdot)\)
\(\chi_{7260}(1993,\cdot)\)
\(\chi_{7260}(2053,\cdot)\)
\(\chi_{7260}(2173,\cdot)\)
\(\chi_{7260}(2197,\cdot)\)
\(\chi_{7260}(2257,\cdot)\)
\(\chi_{7260}(2317,\cdot)\)
\(\chi_{7260}(2437,\cdot)\)
\(\chi_{7260}(2593,\cdot)\)
...
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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 ]
\((3631,4841,4357,7141)\) → \((1,1,-i,e\left(\frac{101}{110}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(7\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 7260 }(13, a) \) |
\(1\) | \(1\) | \(e\left(\frac{39}{220}\right)\) | \(e\left(\frac{217}{220}\right)\) | \(e\left(\frac{163}{220}\right)\) | \(e\left(\frac{39}{55}\right)\) | \(e\left(\frac{23}{44}\right)\) | \(e\left(\frac{6}{55}\right)\) | \(e\left(\frac{53}{55}\right)\) | \(e\left(\frac{69}{220}\right)\) | \(e\left(\frac{13}{110}\right)\) | \(e\left(\frac{9}{44}\right)\) |
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sage:chi.jacobi_sum(n)
