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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7581, base_ring=CyclotomicField(114))
M = H._module
chi = DirichletCharacter(H, M([0,38,82]))
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pari:[g,chi] = znchar(Mod(5749,7581))
\(\chi_{7581}(163,\cdot)\)
\(\chi_{7581}(235,\cdot)\)
\(\chi_{7581}(562,\cdot)\)
\(\chi_{7581}(634,\cdot)\)
\(\chi_{7581}(961,\cdot)\)
\(\chi_{7581}(1033,\cdot)\)
\(\chi_{7581}(1360,\cdot)\)
\(\chi_{7581}(1432,\cdot)\)
\(\chi_{7581}(1759,\cdot)\)
\(\chi_{7581}(1831,\cdot)\)
\(\chi_{7581}(2158,\cdot)\)
\(\chi_{7581}(2230,\cdot)\)
\(\chi_{7581}(2557,\cdot)\)
\(\chi_{7581}(2629,\cdot)\)
\(\chi_{7581}(3028,\cdot)\)
\(\chi_{7581}(3355,\cdot)\)
\(\chi_{7581}(3427,\cdot)\)
\(\chi_{7581}(3754,\cdot)\)
\(\chi_{7581}(3826,\cdot)\)
\(\chi_{7581}(4153,\cdot)\)
\(\chi_{7581}(4225,\cdot)\)
\(\chi_{7581}(4552,\cdot)\)
\(\chi_{7581}(4951,\cdot)\)
\(\chi_{7581}(5023,\cdot)\)
\(\chi_{7581}(5350,\cdot)\)
\(\chi_{7581}(5422,\cdot)\)
\(\chi_{7581}(5749,\cdot)\)
\(\chi_{7581}(5821,\cdot)\)
\(\chi_{7581}(6148,\cdot)\)
\(\chi_{7581}(6220,\cdot)\)
...
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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 ]
\((2528,6499,1807)\) → \((1,e\left(\frac{1}{3}\right),e\left(\frac{41}{57}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(13\) | \(16\) | \(17\) | \(20\) |
| \( \chi_{ 7581 }(5749, a) \) |
\(1\) | \(1\) | \(e\left(\frac{22}{57}\right)\) | \(e\left(\frac{44}{57}\right)\) | \(e\left(\frac{31}{57}\right)\) | \(e\left(\frac{3}{19}\right)\) | \(e\left(\frac{53}{57}\right)\) | \(e\left(\frac{40}{57}\right)\) | \(e\left(\frac{37}{57}\right)\) | \(e\left(\frac{31}{57}\right)\) | \(e\left(\frac{17}{19}\right)\) | \(e\left(\frac{6}{19}\right)\) |
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sage:chi.jacobi_sum(n)
