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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(40310, base_ring=CyclotomicField(1932))
M = H._module
chi = DirichletCharacter(H, M([1449,552,658]))
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pari:[g,chi] = znchar(Mod(53,40310))
\(\chi_{40310}(53,\cdot)\)
\(\chi_{40310}(123,\cdot)\)
\(\chi_{40310}(197,\cdot)\)
\(\chi_{40310}(227,\cdot)\)
\(\chi_{40310}(297,\cdot)\)
\(\chi_{40310}(393,\cdot)\)
\(\chi_{40310}(397,\cdot)\)
\(\chi_{40310}(413,\cdot)\)
\(\chi_{40310}(487,\cdot)\)
\(\chi_{40310}(547,\cdot)\)
\(\chi_{40310}(803,\cdot)\)
\(\chi_{40310}(837,\cdot)\)
\(\chi_{40310}(953,\cdot)\)
\(\chi_{40310}(1127,\cdot)\)
\(\chi_{40310}(1213,\cdot)\)
\(\chi_{40310}(1263,\cdot)\)
\(\chi_{40310}(1283,\cdot)\)
\(\chi_{40310}(1383,\cdot)\)
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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 ]
\((24187,19461,16821)\) → \((-i,e\left(\frac{2}{7}\right),e\left(\frac{47}{138}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
| \( \chi_{ 40310 }(53, a) \) |
\(1\) | \(1\) | \(e\left(\frac{1241}{1932}\right)\) | \(e\left(\frac{401}{1932}\right)\) | \(e\left(\frac{275}{966}\right)\) | \(e\left(\frac{13}{483}\right)\) | \(e\left(\frac{367}{1932}\right)\) | \(e\left(\frac{53}{276}\right)\) | \(e\left(\frac{409}{483}\right)\) | \(e\left(\frac{821}{966}\right)\) | \(e\left(\frac{103}{644}\right)\) | \(e\left(\frac{597}{644}\right)\) |
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sage:chi.jacobi_sum(n)
