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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6069, base_ring=CyclotomicField(816))
M = H._module
chi = DirichletCharacter(H, M([0,136,27]))
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pari:[g,chi] = znchar(Mod(31,6069))
\(\chi_{6069}(10,\cdot)\)
\(\chi_{6069}(31,\cdot)\)
\(\chi_{6069}(61,\cdot)\)
\(\chi_{6069}(73,\cdot)\)
\(\chi_{6069}(82,\cdot)\)
\(\chi_{6069}(124,\cdot)\)
\(\chi_{6069}(199,\cdot)\)
\(\chi_{6069}(241,\cdot)\)
\(\chi_{6069}(250,\cdot)\)
\(\chi_{6069}(262,\cdot)\)
\(\chi_{6069}(283,\cdot)\)
\(\chi_{6069}(292,\cdot)\)
\(\chi_{6069}(313,\cdot)\)
\(\chi_{6069}(334,\cdot)\)
\(\chi_{6069}(346,\cdot)\)
\(\chi_{6069}(367,\cdot)\)
\(\chi_{6069}(388,\cdot)\)
\(\chi_{6069}(397,\cdot)\)
\(\chi_{6069}(418,\cdot)\)
\(\chi_{6069}(430,\cdot)\)
\(\chi_{6069}(439,\cdot)\)
\(\chi_{6069}(481,\cdot)\)
\(\chi_{6069}(556,\cdot)\)
\(\chi_{6069}(598,\cdot)\)
\(\chi_{6069}(607,\cdot)\)
\(\chi_{6069}(619,\cdot)\)
\(\chi_{6069}(640,\cdot)\)
\(\chi_{6069}(649,\cdot)\)
\(\chi_{6069}(670,\cdot)\)
\(\chi_{6069}(691,\cdot)\)
...
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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 ]
\((2024,4336,3760)\) → \((1,e\left(\frac{1}{6}\right),e\left(\frac{9}{272}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(13\) | \(16\) | \(19\) | \(20\) |
| \( \chi_{ 6069 }(31, a) \) |
\(1\) | \(1\) | \(e\left(\frac{253}{408}\right)\) | \(e\left(\frac{49}{204}\right)\) | \(e\left(\frac{335}{816}\right)\) | \(e\left(\frac{117}{136}\right)\) | \(e\left(\frac{25}{816}\right)\) | \(e\left(\frac{349}{816}\right)\) | \(e\left(\frac{67}{68}\right)\) | \(e\left(\frac{49}{102}\right)\) | \(e\left(\frac{121}{408}\right)\) | \(e\left(\frac{177}{272}\right)\) |
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sage:chi.jacobi_sum(n)
