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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5175, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([22,33,24]))
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pari:[g,chi] = znchar(Mod(593,5175))
\(\chi_{5175}(593,\cdot)\)
\(\chi_{5175}(818,\cdot)\)
\(\chi_{5175}(1007,\cdot)\)
\(\chi_{5175}(1043,\cdot)\)
\(\chi_{5175}(1232,\cdot)\)
\(\chi_{5175}(1268,\cdot)\)
\(\chi_{5175}(1457,\cdot)\)
\(\chi_{5175}(1682,\cdot)\)
\(\chi_{5175}(1718,\cdot)\)
\(\chi_{5175}(2132,\cdot)\)
\(\chi_{5175}(2168,\cdot)\)
\(\chi_{5175}(2582,\cdot)\)
\(\chi_{5175}(3068,\cdot)\)
\(\chi_{5175}(3293,\cdot)\)
\(\chi_{5175}(3482,\cdot)\)
\(\chi_{5175}(3707,\cdot)\)
\(\chi_{5175}(3968,\cdot)\)
\(\chi_{5175}(4382,\cdot)\)
\(\chi_{5175}(4418,\cdot)\)
\(\chi_{5175}(4832,\cdot)\)
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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 ]
\((4601,3727,2926)\) → \((-1,-i,e\left(\frac{6}{11}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 5175 }(593, a) \) |
\(1\) | \(1\) | \(e\left(\frac{15}{44}\right)\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{5}{44}\right)\) | \(e\left(\frac{1}{44}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{39}{44}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{3}{44}\right)\) | \(e\left(\frac{15}{22}\right)\) |
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sage:chi.jacobi_sum(n)
