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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1380, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([22,0,11,4]))
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pari:[g,chi] = znchar(Mod(1267,1380))
\(\chi_{1380}(127,\cdot)\)
\(\chi_{1380}(163,\cdot)\)
\(\chi_{1380}(187,\cdot)\)
\(\chi_{1380}(223,\cdot)\)
\(\chi_{1380}(307,\cdot)\)
\(\chi_{1380}(403,\cdot)\)
\(\chi_{1380}(427,\cdot)\)
\(\chi_{1380}(463,\cdot)\)
\(\chi_{1380}(487,\cdot)\)
\(\chi_{1380}(547,\cdot)\)
\(\chi_{1380}(583,\cdot)\)
\(\chi_{1380}(607,\cdot)\)
\(\chi_{1380}(703,\cdot)\)
\(\chi_{1380}(763,\cdot)\)
\(\chi_{1380}(823,\cdot)\)
\(\chi_{1380}(883,\cdot)\)
\(\chi_{1380}(1087,\cdot)\)
\(\chi_{1380}(1267,\cdot)\)
\(\chi_{1380}(1327,\cdot)\)
\(\chi_{1380}(1363,\cdot)\)
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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 ]
\((691,461,277,1201)\) → \((-1,1,i,e\left(\frac{1}{11}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 1380 }(1267, a) \) |
\(1\) | \(1\) | \(e\left(\frac{21}{44}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{1}{44}\right)\) | \(e\left(\frac{39}{44}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{7}{44}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{31}{44}\right)\) |
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sage:chi.jacobi_sum(n)
