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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1050, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([0,57,40]))
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pari:[g,chi] = znchar(Mod(613,1050))
\(\chi_{1050}(37,\cdot)\)
\(\chi_{1050}(67,\cdot)\)
\(\chi_{1050}(163,\cdot)\)
\(\chi_{1050}(247,\cdot)\)
\(\chi_{1050}(277,\cdot)\)
\(\chi_{1050}(373,\cdot)\)
\(\chi_{1050}(403,\cdot)\)
\(\chi_{1050}(487,\cdot)\)
\(\chi_{1050}(583,\cdot)\)
\(\chi_{1050}(613,\cdot)\)
\(\chi_{1050}(667,\cdot)\)
\(\chi_{1050}(697,\cdot)\)
\(\chi_{1050}(823,\cdot)\)
\(\chi_{1050}(877,\cdot)\)
\(\chi_{1050}(1003,\cdot)\)
\(\chi_{1050}(1033,\cdot)\)
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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 ]
\((701,127,451)\) → \((1,e\left(\frac{19}{20}\right),e\left(\frac{2}{3}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 1050 }(613, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{1}{60}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{47}{60}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(i\) |
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sage:chi.jacobi_sum(n)
