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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(966, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([0,11,45]))
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pari:[g,chi] = znchar(Mod(157,966))
\(\chi_{966}(19,\cdot)\)
\(\chi_{966}(61,\cdot)\)
\(\chi_{966}(103,\cdot)\)
\(\chi_{966}(145,\cdot)\)
\(\chi_{966}(157,\cdot)\)
\(\chi_{966}(199,\cdot)\)
\(\chi_{966}(241,\cdot)\)
\(\chi_{966}(283,\cdot)\)
\(\chi_{966}(313,\cdot)\)
\(\chi_{966}(355,\cdot)\)
\(\chi_{966}(451,\cdot)\)
\(\chi_{966}(481,\cdot)\)
\(\chi_{966}(493,\cdot)\)
\(\chi_{966}(523,\cdot)\)
\(\chi_{966}(619,\cdot)\)
\(\chi_{966}(649,\cdot)\)
\(\chi_{966}(661,\cdot)\)
\(\chi_{966}(733,\cdot)\)
\(\chi_{966}(787,\cdot)\)
\(\chi_{966}(871,\cdot)\)
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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 ]
\((323,829,925)\) → \((1,e\left(\frac{1}{6}\right),e\left(\frac{15}{22}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(25\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 966 }(157, a) \) |
\(1\) | \(1\) | \(e\left(\frac{17}{33}\right)\) | \(e\left(\frac{53}{66}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{31}{33}\right)\) | \(e\left(\frac{2}{33}\right)\) | \(e\left(\frac{1}{33}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{17}{66}\right)\) | \(e\left(\frac{43}{66}\right)\) | \(e\left(\frac{15}{22}\right)\) |
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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)
