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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3234, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([21,25,0]))
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pari:[g,chi] = znchar(Mod(2663,3234))
\(\chi_{3234}(89,\cdot)\)
\(\chi_{3234}(353,\cdot)\)
\(\chi_{3234}(551,\cdot)\)
\(\chi_{3234}(1013,\cdot)\)
\(\chi_{3234}(1277,\cdot)\)
\(\chi_{3234}(1475,\cdot)\)
\(\chi_{3234}(1739,\cdot)\)
\(\chi_{3234}(1937,\cdot)\)
\(\chi_{3234}(2201,\cdot)\)
\(\chi_{3234}(2399,\cdot)\)
\(\chi_{3234}(2663,\cdot)\)
\(\chi_{3234}(3125,\cdot)\)
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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 ]
\((1079,199,2059)\) → \((-1,e\left(\frac{25}{42}\right),1)\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 3234 }(2663, a) \) |
\(1\) | \(1\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{3}{7}\right)\) |
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sage:chi.jacobi_sum(n)
