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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(805, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([33,0,12]))
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pari:[g,chi] = znchar(Mod(8,805))
\(\chi_{805}(8,\cdot)\)
\(\chi_{805}(78,\cdot)\)
\(\chi_{805}(127,\cdot)\)
\(\chi_{805}(197,\cdot)\)
\(\chi_{805}(232,\cdot)\)
\(\chi_{805}(288,\cdot)\)
\(\chi_{805}(302,\cdot)\)
\(\chi_{805}(358,\cdot)\)
\(\chi_{805}(372,\cdot)\)
\(\chi_{805}(393,\cdot)\)
\(\chi_{805}(407,\cdot)\)
\(\chi_{805}(463,\cdot)\)
\(\chi_{805}(512,\cdot)\)
\(\chi_{805}(533,\cdot)\)
\(\chi_{805}(547,\cdot)\)
\(\chi_{805}(568,\cdot)\)
\(\chi_{805}(652,\cdot)\)
\(\chi_{805}(673,\cdot)\)
\(\chi_{805}(708,\cdot)\)
\(\chi_{805}(722,\cdot)\)
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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 ]
\((162,346,281)\) → \((-i,1,e\left(\frac{3}{11}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) | \(16\) |
| \( \chi_{ 805 }(8, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{13}{44}\right)\) | \(e\left(\frac{27}{44}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{39}{44}\right)\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{9}{44}\right)\) | \(e\left(\frac{3}{44}\right)\) | \(e\left(\frac{2}{11}\right)\) |
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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)
