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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(30345, base_ring=CyclotomicField(272))
M = H._module
chi = DirichletCharacter(H, M([0,204,0,233]))
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pari:[g,chi] = znchar(Mod(6763,30345))
\(\chi_{30345}(148,\cdot)\)
\(\chi_{30345}(232,\cdot)\)
\(\chi_{30345}(652,\cdot)\)
\(\chi_{30345}(1093,\cdot)\)
\(\chi_{30345}(1282,\cdot)\)
\(\chi_{30345}(1387,\cdot)\)
\(\chi_{30345}(1408,\cdot)\)
\(\chi_{30345}(1618,\cdot)\)
\(\chi_{30345}(1933,\cdot)\)
\(\chi_{30345}(2017,\cdot)\)
\(\chi_{30345}(2437,\cdot)\)
\(\chi_{30345}(2878,\cdot)\)
\(\chi_{30345}(3067,\cdot)\)
\(\chi_{30345}(3172,\cdot)\)
\(\chi_{30345}(3193,\cdot)\)
\(\chi_{30345}(3718,\cdot)\)
\(\chi_{30345}(3802,\cdot)\)
\(\chi_{30345}(4222,\cdot)\)
\(\chi_{30345}(4663,\cdot)\)
\(\chi_{30345}(4852,\cdot)\)
\(\chi_{30345}(4957,\cdot)\)
\(\chi_{30345}(5188,\cdot)\)
\(\chi_{30345}(5503,\cdot)\)
\(\chi_{30345}(5587,\cdot)\)
\(\chi_{30345}(6007,\cdot)\)
\(\chi_{30345}(6448,\cdot)\)
\(\chi_{30345}(6637,\cdot)\)
\(\chi_{30345}(6742,\cdot)\)
\(\chi_{30345}(6763,\cdot)\)
\(\chi_{30345}(6973,\cdot)\)
...
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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 ]
\((20231,24277,4336,28036)\) → \((1,-i,1,e\left(\frac{233}{272}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(8\) | \(11\) | \(13\) | \(16\) | \(19\) | \(22\) | \(23\) | \(26\) |
| \( \chi_{ 30345 }(6763, a) \) |
\(1\) | \(1\) | \(e\left(\frac{69}{136}\right)\) | \(e\left(\frac{1}{68}\right)\) | \(e\left(\frac{71}{136}\right)\) | \(e\left(\frac{191}{272}\right)\) | \(e\left(\frac{5}{34}\right)\) | \(e\left(\frac{1}{34}\right)\) | \(e\left(\frac{67}{136}\right)\) | \(e\left(\frac{57}{272}\right)\) | \(e\left(\frac{75}{272}\right)\) | \(e\left(\frac{89}{136}\right)\) |
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sage:chi.jacobi_sum(n)
