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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(40310, base_ring=CyclotomicField(644))
M = H._module
chi = DirichletCharacter(H, M([483,138,238]))
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pari:[g,chi] = znchar(Mod(1543,40310))
\(\chi_{40310}(33,\cdot)\)
\(\chi_{40310}(187,\cdot)\)
\(\chi_{40310}(353,\cdot)\)
\(\chi_{40310}(383,\cdot)\)
\(\chi_{40310}(477,\cdot)\)
\(\chi_{40310}(643,\cdot)\)
\(\chi_{40310}(937,\cdot)\)
\(\chi_{40310}(1057,\cdot)\)
\(\chi_{40310}(1327,\cdot)\)
\(\chi_{40310}(1543,\cdot)\)
\(\chi_{40310}(1773,\cdot)\)
\(\chi_{40310}(2093,\cdot)\)
\(\chi_{40310}(2703,\cdot)\)
\(\chi_{40310}(2913,\cdot)\)
\(\chi_{40310}(2933,\cdot)\)
\(\chi_{40310}(2967,\cdot)\)
\(\chi_{40310}(2993,\cdot)\)
\(\chi_{40310}(3257,\cdot)\)
\(\chi_{40310}(3647,\cdot)\)
\(\chi_{40310}(3717,\cdot)\)
\(\chi_{40310}(3747,\cdot)\)
\(\chi_{40310}(3763,\cdot)\)
\(\chi_{40310}(3837,\cdot)\)
\(\chi_{40310}(4093,\cdot)\)
\(\chi_{40310}(4383,\cdot)\)
\(\chi_{40310}(5137,\cdot)\)
\(\chi_{40310}(5387,\cdot)\)
\(\chi_{40310}(5503,\cdot)\)
\(\chi_{40310}(5693,\cdot)\)
\(\chi_{40310}(5747,\cdot)\)
...
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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 ]
\((24187,19461,16821)\) → \((-i,e\left(\frac{3}{14}\right),e\left(\frac{17}{46}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
| \( \chi_{ 40310 }(1543, a) \) |
\(1\) | \(1\) | \(e\left(\frac{305}{644}\right)\) | \(e\left(\frac{515}{644}\right)\) | \(e\left(\frac{305}{322}\right)\) | \(e\left(\frac{143}{322}\right)\) | \(e\left(\frac{489}{644}\right)\) | \(e\left(\frac{73}{92}\right)\) | \(e\left(\frac{313}{322}\right)\) | \(e\left(\frac{44}{161}\right)\) | \(e\left(\frac{331}{644}\right)\) | \(e\left(\frac{271}{644}\right)\) |
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sage:chi.jacobi_sum(n)
