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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1860, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([0,30,45,4]))
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pari:[g,chi] = znchar(Mod(1373,1860))
\(\chi_{1860}(113,\cdot)\)
\(\chi_{1860}(173,\cdot)\)
\(\chi_{1860}(257,\cdot)\)
\(\chi_{1860}(293,\cdot)\)
\(\chi_{1860}(317,\cdot)\)
\(\chi_{1860}(413,\cdot)\)
\(\chi_{1860}(617,\cdot)\)
\(\chi_{1860}(857,\cdot)\)
\(\chi_{1860}(917,\cdot)\)
\(\chi_{1860}(1037,\cdot)\)
\(\chi_{1860}(1073,\cdot)\)
\(\chi_{1860}(1157,\cdot)\)
\(\chi_{1860}(1373,\cdot)\)
\(\chi_{1860}(1433,\cdot)\)
\(\chi_{1860}(1733,\cdot)\)
\(\chi_{1860}(1817,\cdot)\)
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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 ]
\((931,1241,1117,1801)\) → \((1,-1,-i,e\left(\frac{1}{15}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 1860 }(1373, a) \) |
\(1\) | \(1\) | \(e\left(\frac{37}{60}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{59}{60}\right)\) | \(e\left(\frac{43}{60}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{31}{60}\right)\) |
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sage:chi.jacobi_sum(n)
