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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5175, base_ring=CyclotomicField(220))
M = H._module
chi = DirichletCharacter(H, M([0,143,10]))
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pari:[g,chi] = znchar(Mod(442,5175))
\(\chi_{5175}(28,\cdot)\)
\(\chi_{5175}(37,\cdot)\)
\(\chi_{5175}(172,\cdot)\)
\(\chi_{5175}(217,\cdot)\)
\(\chi_{5175}(352,\cdot)\)
\(\chi_{5175}(388,\cdot)\)
\(\chi_{5175}(433,\cdot)\)
\(\chi_{5175}(442,\cdot)\)
\(\chi_{5175}(523,\cdot)\)
\(\chi_{5175}(613,\cdot)\)
\(\chi_{5175}(658,\cdot)\)
\(\chi_{5175}(802,\cdot)\)
\(\chi_{5175}(838,\cdot)\)
\(\chi_{5175}(847,\cdot)\)
\(\chi_{5175}(937,\cdot)\)
\(\chi_{5175}(973,\cdot)\)
\(\chi_{5175}(1027,\cdot)\)
\(\chi_{5175}(1063,\cdot)\)
\(\chi_{5175}(1072,\cdot)\)
\(\chi_{5175}(1252,\cdot)\)
\(\chi_{5175}(1378,\cdot)\)
\(\chi_{5175}(1387,\cdot)\)
\(\chi_{5175}(1423,\cdot)\)
\(\chi_{5175}(1477,\cdot)\)
\(\chi_{5175}(1558,\cdot)\)
\(\chi_{5175}(1648,\cdot)\)
\(\chi_{5175}(1792,\cdot)\)
\(\chi_{5175}(1828,\cdot)\)
\(\chi_{5175}(1837,\cdot)\)
\(\chi_{5175}(1873,\cdot)\)
...
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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 ]
\((4601,3727,2926)\) → \((1,e\left(\frac{13}{20}\right),e\left(\frac{1}{22}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 5175 }(442, a) \) |
\(1\) | \(1\) | \(e\left(\frac{163}{220}\right)\) | \(e\left(\frac{53}{110}\right)\) | \(e\left(\frac{5}{44}\right)\) | \(e\left(\frac{49}{220}\right)\) | \(e\left(\frac{89}{110}\right)\) | \(e\left(\frac{217}{220}\right)\) | \(e\left(\frac{47}{55}\right)\) | \(e\left(\frac{53}{55}\right)\) | \(e\left(\frac{169}{220}\right)\) | \(e\left(\frac{21}{55}\right)\) |
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sage:chi.jacobi_sum(n)
