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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8732, base_ring=CyclotomicField(174))
M = H._module
chi = DirichletCharacter(H, M([0,116,120]))
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pari:[g,chi] = znchar(Mod(4265,8732))
\(\chi_{8732}(121,\cdot)\)
\(\chi_{8732}(137,\cdot)\)
\(\chi_{8732}(285,\cdot)\)
\(\chi_{8732}(417,\cdot)\)
\(\chi_{8732}(433,\cdot)\)
\(\chi_{8732}(713,\cdot)\)
\(\chi_{8732}(729,\cdot)\)
\(\chi_{8732}(861,\cdot)\)
\(\chi_{8732}(877,\cdot)\)
\(\chi_{8732}(1025,\cdot)\)
\(\chi_{8732}(1157,\cdot)\)
\(\chi_{8732}(1305,\cdot)\)
\(\chi_{8732}(1469,\cdot)\)
\(\chi_{8732}(1897,\cdot)\)
\(\chi_{8732}(1913,\cdot)\)
\(\chi_{8732}(2209,\cdot)\)
\(\chi_{8732}(2505,\cdot)\)
\(\chi_{8732}(2637,\cdot)\)
\(\chi_{8732}(2653,\cdot)\)
\(\chi_{8732}(2785,\cdot)\)
\(\chi_{8732}(2801,\cdot)\)
\(\chi_{8732}(3097,\cdot)\)
\(\chi_{8732}(3673,\cdot)\)
\(\chi_{8732}(3821,\cdot)\)
\(\chi_{8732}(3969,\cdot)\)
\(\chi_{8732}(4117,\cdot)\)
\(\chi_{8732}(4133,\cdot)\)
\(\chi_{8732}(4265,\cdot)\)
\(\chi_{8732}(4429,\cdot)\)
\(\chi_{8732}(4709,\cdot)\)
...
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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 ]
\((4367,1889,297)\) → \((1,e\left(\frac{2}{3}\right),e\left(\frac{20}{29}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 8732 }(4265, a) \) |
\(1\) | \(1\) | \(e\left(\frac{71}{87}\right)\) | \(e\left(\frac{41}{87}\right)\) | \(e\left(\frac{65}{87}\right)\) | \(e\left(\frac{55}{87}\right)\) | \(e\left(\frac{7}{29}\right)\) | \(e\left(\frac{32}{87}\right)\) | \(e\left(\frac{25}{87}\right)\) | \(e\left(\frac{22}{87}\right)\) | \(e\left(\frac{47}{87}\right)\) | \(e\left(\frac{49}{87}\right)\) |
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sage:chi.jacobi_sum(n)
