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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(552, base_ring=CyclotomicField(22))
M = H._module
chi = DirichletCharacter(H, M([0,11,0,10]))
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pari:[g,chi] = znchar(Mod(469,552))
\(\chi_{552}(13,\cdot)\)
\(\chi_{552}(85,\cdot)\)
\(\chi_{552}(133,\cdot)\)
\(\chi_{552}(301,\cdot)\)
\(\chi_{552}(325,\cdot)\)
\(\chi_{552}(349,\cdot)\)
\(\chi_{552}(397,\cdot)\)
\(\chi_{552}(445,\cdot)\)
\(\chi_{552}(469,\cdot)\)
\(\chi_{552}(541,\cdot)\)
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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 ]
\((415,277,185,97)\) → \((1,-1,1,e\left(\frac{5}{11}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 552 }(469, a) \) |
\(1\) | \(1\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{13}{22}\right)\) |
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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)
