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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([57,0,7]))
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pari:[g,chi] = znchar(Mod(4796,7581))
\(\chi_{7581}(8,\cdot)\)
\(\chi_{7581}(50,\cdot)\)
\(\chi_{7581}(407,\cdot)\)
\(\chi_{7581}(449,\cdot)\)
\(\chi_{7581}(806,\cdot)\)
\(\chi_{7581}(848,\cdot)\)
\(\chi_{7581}(1205,\cdot)\)
\(\chi_{7581}(1247,\cdot)\)
\(\chi_{7581}(1604,\cdot)\)
\(\chi_{7581}(1646,\cdot)\)
\(\chi_{7581}(2003,\cdot)\)
\(\chi_{7581}(2045,\cdot)\)
\(\chi_{7581}(2402,\cdot)\)
\(\chi_{7581}(2444,\cdot)\)
\(\chi_{7581}(2801,\cdot)\)
\(\chi_{7581}(2843,\cdot)\)
\(\chi_{7581}(3200,\cdot)\)
\(\chi_{7581}(3242,\cdot)\)
\(\chi_{7581}(3599,\cdot)\)
\(\chi_{7581}(3641,\cdot)\)
\(\chi_{7581}(3998,\cdot)\)
\(\chi_{7581}(4397,\cdot)\)
\(\chi_{7581}(4439,\cdot)\)
\(\chi_{7581}(4796,\cdot)\)
\(\chi_{7581}(4838,\cdot)\)
\(\chi_{7581}(5195,\cdot)\)
\(\chi_{7581}(5237,\cdot)\)
\(\chi_{7581}(5594,\cdot)\)
\(\chi_{7581}(5636,\cdot)\)
\(\chi_{7581}(5993,\cdot)\)
...
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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 ]
\((2528,6499,1807)\) → \((-1,1,e\left(\frac{7}{114}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(13\) | \(16\) | \(17\) | \(20\) |
| \( \chi_{ 7581 }(4796, a) \) |
\(1\) | \(1\) | \(e\left(\frac{32}{57}\right)\) | \(e\left(\frac{7}{57}\right)\) | \(e\left(\frac{85}{114}\right)\) | \(e\left(\frac{13}{19}\right)\) | \(e\left(\frac{35}{114}\right)\) | \(e\left(\frac{29}{38}\right)\) | \(e\left(\frac{23}{114}\right)\) | \(e\left(\frac{14}{57}\right)\) | \(e\left(\frac{43}{114}\right)\) | \(e\left(\frac{33}{38}\right)\) |
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sage:chi.jacobi_sum(n)
