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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8464, base_ring=CyclotomicField(22))
M = H._module
chi = DirichletCharacter(H, M([11,11,9]))
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pari:[g,chi] = znchar(Mod(2311,8464))
\(\chi_{8464}(263,\cdot)\)
\(\chi_{8464}(359,\cdot)\)
\(\chi_{8464}(2311,\cdot)\)
\(\chi_{8464}(2775,\cdot)\)
\(\chi_{8464}(2919,\cdot)\)
\(\chi_{8464}(4055,\cdot)\)
\(\chi_{8464}(4295,\cdot)\)
\(\chi_{8464}(5847,\cdot)\)
\(\chi_{8464}(6759,\cdot)\)
\(\chi_{8464}(6919,\cdot)\)
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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 ]
\((7407,2117,6353)\) → \((-1,-1,e\left(\frac{9}{22}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 8464 }(2311, a) \) |
\(1\) | \(1\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{9}{11}\right)\) |
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sage:chi.jacobi_sum(n)
