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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(40310, base_ring=CyclotomicField(966))
M = H._module
chi = DirichletCharacter(H, M([483,345,574]))
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pari:[g,chi] = znchar(Mod(9,40310))
\(\chi_{40310}(9,\cdot)\)
\(\chi_{40310}(399,\cdot)\)
\(\chi_{40310}(1019,\cdot)\)
\(\chi_{40310}(1289,\cdot)\)
\(\chi_{40310}(1369,\cdot)\)
\(\chi_{40310}(1459,\cdot)\)
\(\chi_{40310}(1559,\cdot)\)
\(\chi_{40310}(1749,\cdot)\)
\(\chi_{40310}(2139,\cdot)\)
\(\chi_{40310}(2209,\cdot)\)
\(\chi_{40310}(2429,\cdot)\)
\(\chi_{40310}(2449,\cdot)\)
\(\chi_{40310}(2499,\cdot)\)
\(\chi_{40310}(2619,\cdot)\)
\(\chi_{40310}(2719,\cdot)\)
\(\chi_{40310}(2759,\cdot)\)
\(\chi_{40310}(2789,\cdot)\)
\(\chi_{40310}(3109,\cdot)\)
\(\chi_{40310}(3319,\cdot)\)
\(\chi_{40310}(3899,\cdot)\)
\(\chi_{40310}(4069,\cdot)\)
\(\chi_{40310}(4239,\cdot)\)
\(\chi_{40310}(4269,\cdot)\)
\(\chi_{40310}(4459,\cdot)\)
\(\chi_{40310}(4479,\cdot)\)
\(\chi_{40310}(4499,\cdot)\)
\(\chi_{40310}(4529,\cdot)\)
\(\chi_{40310}(5039,\cdot)\)
\(\chi_{40310}(5229,\cdot)\)
\(\chi_{40310}(5329,\cdot)\)
...
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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 ]
\((24187,19461,16821)\) → \((-1,e\left(\frac{5}{14}\right),e\left(\frac{41}{69}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
| \( \chi_{ 40310 }(9, a) \) |
\(1\) | \(1\) | \(e\left(\frac{313}{483}\right)\) | \(e\left(\frac{479}{966}\right)\) | \(e\left(\frac{143}{483}\right)\) | \(e\left(\frac{85}{966}\right)\) | \(e\left(\frac{925}{966}\right)\) | \(e\left(\frac{40}{69}\right)\) | \(e\left(\frac{445}{966}\right)\) | \(e\left(\frac{139}{966}\right)\) | \(e\left(\frac{221}{322}\right)\) | \(e\left(\frac{152}{161}\right)\) |
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sage:chi.jacobi_sum(n)
