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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7865, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([0,30,11]))
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pari:[g,chi] = znchar(Mod(1486,7865))
\(\chi_{7865}(56,\cdot)\)
\(\chi_{7865}(166,\cdot)\)
\(\chi_{7865}(771,\cdot)\)
\(\chi_{7865}(881,\cdot)\)
\(\chi_{7865}(1486,\cdot)\)
\(\chi_{7865}(1596,\cdot)\)
\(\chi_{7865}(2201,\cdot)\)
\(\chi_{7865}(2311,\cdot)\)
\(\chi_{7865}(2916,\cdot)\)
\(\chi_{7865}(3741,\cdot)\)
\(\chi_{7865}(4346,\cdot)\)
\(\chi_{7865}(4456,\cdot)\)
\(\chi_{7865}(5061,\cdot)\)
\(\chi_{7865}(5171,\cdot)\)
\(\chi_{7865}(5776,\cdot)\)
\(\chi_{7865}(5886,\cdot)\)
\(\chi_{7865}(6491,\cdot)\)
\(\chi_{7865}(6601,\cdot)\)
\(\chi_{7865}(7206,\cdot)\)
\(\chi_{7865}(7316,\cdot)\)
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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 ]
\((3147,3511,1211)\) → \((1,e\left(\frac{5}{11}\right),e\left(\frac{1}{6}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(12\) | \(14\) | \(16\) |
| \( \chi_{ 7865 }(1486, a) \) |
\(1\) | \(1\) | \(e\left(\frac{41}{66}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{8}{33}\right)\) | \(e\left(\frac{19}{66}\right)\) | \(e\left(\frac{1}{66}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{16}{33}\right)\) |
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sage:chi.jacobi_sum(n)
