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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(345, base_ring=CyclotomicField(22))
M = H._module
chi = DirichletCharacter(H, M([0,11,4]))
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pari:[g,chi] = znchar(Mod(4,345))
\(\chi_{345}(4,\cdot)\)
\(\chi_{345}(49,\cdot)\)
\(\chi_{345}(64,\cdot)\)
\(\chi_{345}(94,\cdot)\)
\(\chi_{345}(124,\cdot)\)
\(\chi_{345}(154,\cdot)\)
\(\chi_{345}(169,\cdot)\)
\(\chi_{345}(259,\cdot)\)
\(\chi_{345}(289,\cdot)\)
\(\chi_{345}(334,\cdot)\)
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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 ]
\((116,277,166)\) → \((1,-1,e\left(\frac{2}{11}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 345 }(4, a) \) |
\(1\) | \(1\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{8}{11}\right)\) |
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sage:chi.jacobi_sum(n)
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sage:chi.gauss_sum(a)
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pari:znchargauss(g,chi,a)
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sage:chi.jacobi_sum(n)
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sage:chi.kloosterman_sum(a,b)
