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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(30345, base_ring=CyclotomicField(102))
M = H._module
chi = DirichletCharacter(H, M([51,0,85,12]))
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pari:[g,chi] = znchar(Mod(341,30345))
\(\chi_{30345}(341,\cdot)\)
\(\chi_{30345}(1361,\cdot)\)
\(\chi_{30345}(2126,\cdot)\)
\(\chi_{30345}(3146,\cdot)\)
\(\chi_{30345}(3911,\cdot)\)
\(\chi_{30345}(4931,\cdot)\)
\(\chi_{30345}(5696,\cdot)\)
\(\chi_{30345}(6716,\cdot)\)
\(\chi_{30345}(7481,\cdot)\)
\(\chi_{30345}(8501,\cdot)\)
\(\chi_{30345}(9266,\cdot)\)
\(\chi_{30345}(10286,\cdot)\)
\(\chi_{30345}(11051,\cdot)\)
\(\chi_{30345}(12071,\cdot)\)
\(\chi_{30345}(12836,\cdot)\)
\(\chi_{30345}(13856,\cdot)\)
\(\chi_{30345}(14621,\cdot)\)
\(\chi_{30345}(15641,\cdot)\)
\(\chi_{30345}(16406,\cdot)\)
\(\chi_{30345}(17426,\cdot)\)
\(\chi_{30345}(18191,\cdot)\)
\(\chi_{30345}(19211,\cdot)\)
\(\chi_{30345}(19976,\cdot)\)
\(\chi_{30345}(20996,\cdot)\)
\(\chi_{30345}(21761,\cdot)\)
\(\chi_{30345}(22781,\cdot)\)
\(\chi_{30345}(23546,\cdot)\)
\(\chi_{30345}(25331,\cdot)\)
\(\chi_{30345}(26351,\cdot)\)
\(\chi_{30345}(27116,\cdot)\)
...
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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 ]
\((20231,24277,4336,28036)\) → \((-1,1,e\left(\frac{5}{6}\right),e\left(\frac{2}{17}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(8\) | \(11\) | \(13\) | \(16\) | \(19\) | \(22\) | \(23\) | \(26\) |
| \( \chi_{ 30345 }(341, a) \) |
\(1\) | \(1\) | \(e\left(\frac{53}{102}\right)\) | \(e\left(\frac{2}{51}\right)\) | \(e\left(\frac{19}{34}\right)\) | \(e\left(\frac{55}{102}\right)\) | \(e\left(\frac{19}{34}\right)\) | \(e\left(\frac{4}{51}\right)\) | \(e\left(\frac{83}{102}\right)\) | \(e\left(\frac{1}{17}\right)\) | \(e\left(\frac{41}{102}\right)\) | \(e\left(\frac{4}{51}\right)\) |
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sage:chi.jacobi_sum(n)
