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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6069, base_ring=CyclotomicField(204))
M = H._module
chi = DirichletCharacter(H, M([0,68,75]))
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pari:[g,chi] = znchar(Mod(625,6069))
\(\chi_{6069}(4,\cdot)\)
\(\chi_{6069}(268,\cdot)\)
\(\chi_{6069}(310,\cdot)\)
\(\chi_{6069}(319,\cdot)\)
\(\chi_{6069}(361,\cdot)\)
\(\chi_{6069}(625,\cdot)\)
\(\chi_{6069}(667,\cdot)\)
\(\chi_{6069}(676,\cdot)\)
\(\chi_{6069}(718,\cdot)\)
\(\chi_{6069}(982,\cdot)\)
\(\chi_{6069}(1024,\cdot)\)
\(\chi_{6069}(1033,\cdot)\)
\(\chi_{6069}(1075,\cdot)\)
\(\chi_{6069}(1339,\cdot)\)
\(\chi_{6069}(1381,\cdot)\)
\(\chi_{6069}(1390,\cdot)\)
\(\chi_{6069}(1432,\cdot)\)
\(\chi_{6069}(1738,\cdot)\)
\(\chi_{6069}(1747,\cdot)\)
\(\chi_{6069}(1789,\cdot)\)
\(\chi_{6069}(2053,\cdot)\)
\(\chi_{6069}(2095,\cdot)\)
\(\chi_{6069}(2104,\cdot)\)
\(\chi_{6069}(2146,\cdot)\)
\(\chi_{6069}(2410,\cdot)\)
\(\chi_{6069}(2452,\cdot)\)
\(\chi_{6069}(2461,\cdot)\)
\(\chi_{6069}(2503,\cdot)\)
\(\chi_{6069}(2767,\cdot)\)
\(\chi_{6069}(2809,\cdot)\)
...
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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 ]
\((2024,4336,3760)\) → \((1,e\left(\frac{1}{3}\right),e\left(\frac{25}{68}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(13\) | \(16\) | \(19\) | \(20\) |
| \( \chi_{ 6069 }(625, a) \) |
\(1\) | \(1\) | \(e\left(\frac{53}{102}\right)\) | \(e\left(\frac{2}{51}\right)\) | \(e\left(\frac{175}{204}\right)\) | \(e\left(\frac{19}{34}\right)\) | \(e\left(\frac{77}{204}\right)\) | \(e\left(\frac{161}{204}\right)\) | \(e\left(\frac{1}{17}\right)\) | \(e\left(\frac{4}{51}\right)\) | \(e\left(\frac{83}{102}\right)\) | \(e\left(\frac{61}{68}\right)\) |
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sage:chi.jacobi_sum(n)
