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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6069, base_ring=CyclotomicField(68))
M = H._module
chi = DirichletCharacter(H, M([0,34,5]))
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pari:[g,chi] = znchar(Mod(2869,6069))
\(\chi_{6069}(13,\cdot)\)
\(\chi_{6069}(55,\cdot)\)
\(\chi_{6069}(370,\cdot)\)
\(\chi_{6069}(412,\cdot)\)
\(\chi_{6069}(727,\cdot)\)
\(\chi_{6069}(769,\cdot)\)
\(\chi_{6069}(1084,\cdot)\)
\(\chi_{6069}(1126,\cdot)\)
\(\chi_{6069}(1441,\cdot)\)
\(\chi_{6069}(1798,\cdot)\)
\(\chi_{6069}(1840,\cdot)\)
\(\chi_{6069}(2155,\cdot)\)
\(\chi_{6069}(2197,\cdot)\)
\(\chi_{6069}(2512,\cdot)\)
\(\chi_{6069}(2554,\cdot)\)
\(\chi_{6069}(2869,\cdot)\)
\(\chi_{6069}(2911,\cdot)\)
\(\chi_{6069}(3226,\cdot)\)
\(\chi_{6069}(3268,\cdot)\)
\(\chi_{6069}(3583,\cdot)\)
\(\chi_{6069}(3625,\cdot)\)
\(\chi_{6069}(3940,\cdot)\)
\(\chi_{6069}(3982,\cdot)\)
\(\chi_{6069}(4339,\cdot)\)
\(\chi_{6069}(4654,\cdot)\)
\(\chi_{6069}(4696,\cdot)\)
\(\chi_{6069}(5011,\cdot)\)
\(\chi_{6069}(5053,\cdot)\)
\(\chi_{6069}(5368,\cdot)\)
\(\chi_{6069}(5410,\cdot)\)
...
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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 ]
\((2024,4336,3760)\) → \((1,-1,e\left(\frac{5}{68}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(13\) | \(16\) | \(19\) | \(20\) |
| \( \chi_{ 6069 }(2869, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{33}{34}\right)\) | \(e\left(\frac{16}{17}\right)\) | \(e\left(\frac{23}{68}\right)\) | \(e\left(\frac{31}{34}\right)\) | \(e\left(\frac{21}{68}\right)\) | \(e\left(\frac{47}{68}\right)\) | \(e\left(\frac{31}{34}\right)\) | \(e\left(\frac{15}{17}\right)\) | \(e\left(\frac{9}{17}\right)\) | \(e\left(\frac{19}{68}\right)\) |
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sage:chi.jacobi_sum(n)
