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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8712, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([33,0,44,21]))
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pari:[g,chi] = znchar(Mod(5191,8712))
\(\chi_{8712}(175,\cdot)\)
\(\chi_{8712}(439,\cdot)\)
\(\chi_{8712}(1231,\cdot)\)
\(\chi_{8712}(1759,\cdot)\)
\(\chi_{8712}(2023,\cdot)\)
\(\chi_{8712}(2551,\cdot)\)
\(\chi_{8712}(2815,\cdot)\)
\(\chi_{8712}(3343,\cdot)\)
\(\chi_{8712}(3607,\cdot)\)
\(\chi_{8712}(4135,\cdot)\)
\(\chi_{8712}(4399,\cdot)\)
\(\chi_{8712}(4927,\cdot)\)
\(\chi_{8712}(5191,\cdot)\)
\(\chi_{8712}(5719,\cdot)\)
\(\chi_{8712}(5983,\cdot)\)
\(\chi_{8712}(6511,\cdot)\)
\(\chi_{8712}(7303,\cdot)\)
\(\chi_{8712}(7567,\cdot)\)
\(\chi_{8712}(8095,\cdot)\)
\(\chi_{8712}(8359,\cdot)\)
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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 ]
\((6535,4357,1937,5689)\) → \((-1,1,e\left(\frac{2}{3}\right),e\left(\frac{7}{22}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 8712 }(5191, a) \) |
\(1\) | \(1\) | \(e\left(\frac{29}{33}\right)\) | \(e\left(\frac{13}{33}\right)\) | \(e\left(\frac{31}{66}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{7}{66}\right)\) | \(e\left(\frac{25}{33}\right)\) | \(e\left(\frac{5}{66}\right)\) | \(e\left(\frac{13}{66}\right)\) | \(e\left(\frac{3}{11}\right)\) |
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sage:chi.jacobi_sum(n)
