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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(30345, base_ring=CyclotomicField(408))
M = H._module
chi = DirichletCharacter(H, M([204,0,68,207]))
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pari:[g,chi] = znchar(Mod(5771,30345))
\(\chi_{30345}(26,\cdot)\)
\(\chi_{30345}(206,\cdot)\)
\(\chi_{30345}(236,\cdot)\)
\(\chi_{30345}(416,\cdot)\)
\(\chi_{30345}(971,\cdot)\)
\(\chi_{30345}(1181,\cdot)\)
\(\chi_{30345}(1256,\cdot)\)
\(\chi_{30345}(1811,\cdot)\)
\(\chi_{30345}(1991,\cdot)\)
\(\chi_{30345}(2021,\cdot)\)
\(\chi_{30345}(2201,\cdot)\)
\(\chi_{30345}(2831,\cdot)\)
\(\chi_{30345}(2966,\cdot)\)
\(\chi_{30345}(3041,\cdot)\)
\(\chi_{30345}(3596,\cdot)\)
\(\chi_{30345}(3776,\cdot)\)
\(\chi_{30345}(3806,\cdot)\)
\(\chi_{30345}(3986,\cdot)\)
\(\chi_{30345}(4541,\cdot)\)
\(\chi_{30345}(4616,\cdot)\)
\(\chi_{30345}(4751,\cdot)\)
\(\chi_{30345}(4826,\cdot)\)
\(\chi_{30345}(5561,\cdot)\)
\(\chi_{30345}(5591,\cdot)\)
\(\chi_{30345}(5771,\cdot)\)
\(\chi_{30345}(6326,\cdot)\)
\(\chi_{30345}(6401,\cdot)\)
\(\chi_{30345}(6536,\cdot)\)
\(\chi_{30345}(6611,\cdot)\)
\(\chi_{30345}(7166,\cdot)\)
...
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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 ]
\((20231,24277,4336,28036)\) → \((-1,1,e\left(\frac{1}{6}\right),e\left(\frac{69}{136}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(8\) | \(11\) | \(13\) | \(16\) | \(19\) | \(22\) | \(23\) | \(26\) |
| \( \chi_{ 30345 }(5771, a) \) |
\(1\) | \(1\) | \(e\left(\frac{47}{204}\right)\) | \(e\left(\frac{47}{102}\right)\) | \(e\left(\frac{47}{68}\right)\) | \(e\left(\frac{341}{408}\right)\) | \(e\left(\frac{16}{17}\right)\) | \(e\left(\frac{47}{51}\right)\) | \(e\left(\frac{191}{204}\right)\) | \(e\left(\frac{9}{136}\right)\) | \(e\left(\frac{397}{408}\right)\) | \(e\left(\frac{35}{204}\right)\) |
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sage:chi.jacobi_sum(n)
