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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4005, base_ring=CyclotomicField(88))
M = H._module
chi = DirichletCharacter(H, M([44,0,27]))
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pari:[g,chi] = znchar(Mod(341,4005))
\(\chi_{4005}(26,\cdot)\)
\(\chi_{4005}(116,\cdot)\)
\(\chi_{4005}(206,\cdot)\)
\(\chi_{4005}(296,\cdot)\)
\(\chi_{4005}(341,\cdot)\)
\(\chi_{4005}(386,\cdot)\)
\(\chi_{4005}(431,\cdot)\)
\(\chi_{4005}(476,\cdot)\)
\(\chi_{4005}(521,\cdot)\)
\(\chi_{4005}(656,\cdot)\)
\(\chi_{4005}(836,\cdot)\)
\(\chi_{4005}(1061,\cdot)\)
\(\chi_{4005}(1106,\cdot)\)
\(\chi_{4005}(1151,\cdot)\)
\(\chi_{4005}(1376,\cdot)\)
\(\chi_{4005}(1421,\cdot)\)
\(\chi_{4005}(1556,\cdot)\)
\(\chi_{4005}(1826,\cdot)\)
\(\chi_{4005}(1961,\cdot)\)
\(\chi_{4005}(2006,\cdot)\)
\(\chi_{4005}(2231,\cdot)\)
\(\chi_{4005}(2276,\cdot)\)
\(\chi_{4005}(2321,\cdot)\)
\(\chi_{4005}(2546,\cdot)\)
\(\chi_{4005}(2726,\cdot)\)
\(\chi_{4005}(2861,\cdot)\)
\(\chi_{4005}(2906,\cdot)\)
\(\chi_{4005}(2951,\cdot)\)
\(\chi_{4005}(2996,\cdot)\)
\(\chi_{4005}(3041,\cdot)\)
...
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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 ]
\((3116,802,181)\) → \((-1,1,e\left(\frac{27}{88}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 4005 }(341, a) \) |
\(1\) | \(1\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{75}{88}\right)\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{5}{88}\right)\) | \(e\left(\frac{23}{88}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{15}{44}\right)\) | \(e\left(\frac{65}{88}\right)\) |
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sage:chi.jacobi_sum(n)
