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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(390400, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([30,45,3,59]))
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pari:[g,chi] = znchar(Mod(142527,390400))
\(\chi_{390400}(63,\cdot)\)
\(\chi_{390400}(13247,\cdot)\)
\(\chi_{390400}(18367,\cdot)\)
\(\chi_{390400}(24383,\cdot)\)
\(\chi_{390400}(82367,\cdot)\)
\(\chi_{390400}(88383,\cdot)\)
\(\chi_{390400}(93503,\cdot)\)
\(\chi_{390400}(106687,\cdot)\)
\(\chi_{390400}(142527,\cdot)\)
\(\chi_{390400}(149823,\cdot)\)
\(\chi_{390400}(205247,\cdot)\)
\(\chi_{390400}(215487,\cdot)\)
\(\chi_{390400}(281663,\cdot)\)
\(\chi_{390400}(291903,\cdot)\)
\(\chi_{390400}(347327,\cdot)\)
\(\chi_{390400}(354623,\cdot)\)
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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 ]
\((106751,176901,171777,76801)\) → \((-1,-i,e\left(\frac{1}{20}\right),e\left(\frac{59}{60}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
| \( \chi_{ 390400 }(142527, a) \) |
\(-1\) | \(1\) | \(1\) | \(e\left(\frac{13}{30}\right)\) | \(1\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(1\) |
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sage:chi.jacobi_sum(n)
