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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(30345, base_ring=CyclotomicField(136))
M = H._module
chi = DirichletCharacter(H, M([68,34,0,105]))
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pari:[g,chi] = znchar(Mod(3137,30345))
\(\chi_{30345}(722,\cdot)\)
\(\chi_{30345}(848,\cdot)\)
\(\chi_{30345}(1352,\cdot)\)
\(\chi_{30345}(1583,\cdot)\)
\(\chi_{30345}(2507,\cdot)\)
\(\chi_{30345}(2633,\cdot)\)
\(\chi_{30345}(3137,\cdot)\)
\(\chi_{30345}(3368,\cdot)\)
\(\chi_{30345}(4292,\cdot)\)
\(\chi_{30345}(4418,\cdot)\)
\(\chi_{30345}(4922,\cdot)\)
\(\chi_{30345}(5153,\cdot)\)
\(\chi_{30345}(6077,\cdot)\)
\(\chi_{30345}(6707,\cdot)\)
\(\chi_{30345}(6938,\cdot)\)
\(\chi_{30345}(7862,\cdot)\)
\(\chi_{30345}(7988,\cdot)\)
\(\chi_{30345}(8492,\cdot)\)
\(\chi_{30345}(8723,\cdot)\)
\(\chi_{30345}(9773,\cdot)\)
\(\chi_{30345}(10277,\cdot)\)
\(\chi_{30345}(10508,\cdot)\)
\(\chi_{30345}(11432,\cdot)\)
\(\chi_{30345}(11558,\cdot)\)
\(\chi_{30345}(12062,\cdot)\)
\(\chi_{30345}(13217,\cdot)\)
\(\chi_{30345}(13343,\cdot)\)
\(\chi_{30345}(13847,\cdot)\)
\(\chi_{30345}(14078,\cdot)\)
\(\chi_{30345}(15002,\cdot)\)
...
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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 ]
\((20231,24277,4336,28036)\) → \((-1,i,1,e\left(\frac{105}{136}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(8\) | \(11\) | \(13\) | \(16\) | \(19\) | \(22\) | \(23\) | \(26\) |
| \( \chi_{ 30345 }(3137, a) \) |
\(1\) | \(1\) | \(e\left(\frac{15}{34}\right)\) | \(e\left(\frac{15}{17}\right)\) | \(e\left(\frac{11}{34}\right)\) | \(e\left(\frac{35}{136}\right)\) | \(e\left(\frac{5}{68}\right)\) | \(e\left(\frac{13}{17}\right)\) | \(e\left(\frac{21}{68}\right)\) | \(e\left(\frac{95}{136}\right)\) | \(e\left(\frac{57}{136}\right)\) | \(e\left(\frac{35}{68}\right)\) |
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sage:chi.jacobi_sum(n)
