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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6069, base_ring=CyclotomicField(102))
M = H._module
chi = DirichletCharacter(H, M([0,17,90]))
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pari:[g,chi] = znchar(Mod(2551,6069))
\(\chi_{6069}(52,\cdot)\)
\(\chi_{6069}(103,\cdot)\)
\(\chi_{6069}(409,\cdot)\)
\(\chi_{6069}(460,\cdot)\)
\(\chi_{6069}(766,\cdot)\)
\(\chi_{6069}(817,\cdot)\)
\(\chi_{6069}(1123,\cdot)\)
\(\chi_{6069}(1174,\cdot)\)
\(\chi_{6069}(1480,\cdot)\)
\(\chi_{6069}(1531,\cdot)\)
\(\chi_{6069}(1837,\cdot)\)
\(\chi_{6069}(1888,\cdot)\)
\(\chi_{6069}(2194,\cdot)\)
\(\chi_{6069}(2245,\cdot)\)
\(\chi_{6069}(2551,\cdot)\)
\(\chi_{6069}(2908,\cdot)\)
\(\chi_{6069}(2959,\cdot)\)
\(\chi_{6069}(3265,\cdot)\)
\(\chi_{6069}(3316,\cdot)\)
\(\chi_{6069}(3622,\cdot)\)
\(\chi_{6069}(3673,\cdot)\)
\(\chi_{6069}(3979,\cdot)\)
\(\chi_{6069}(4030,\cdot)\)
\(\chi_{6069}(4387,\cdot)\)
\(\chi_{6069}(4693,\cdot)\)
\(\chi_{6069}(4744,\cdot)\)
\(\chi_{6069}(5050,\cdot)\)
\(\chi_{6069}(5101,\cdot)\)
\(\chi_{6069}(5407,\cdot)\)
\(\chi_{6069}(5458,\cdot)\)
...
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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 ]
\((2024,4336,3760)\) → \((1,e\left(\frac{1}{6}\right),e\left(\frac{15}{17}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(13\) | \(16\) | \(19\) | \(20\) |
| \( \chi_{ 6069 }(2551, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{50}{51}\right)\) | \(e\left(\frac{49}{51}\right)\) | \(e\left(\frac{91}{102}\right)\) | \(e\left(\frac{16}{17}\right)\) | \(e\left(\frac{89}{102}\right)\) | \(e\left(\frac{49}{51}\right)\) | \(e\left(\frac{15}{34}\right)\) | \(e\left(\frac{47}{51}\right)\) | \(e\left(\frac{19}{102}\right)\) | \(e\left(\frac{29}{34}\right)\) |
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sage:chi.jacobi_sum(n)
