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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(414, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([55,3]))
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pari:[g,chi] = znchar(Mod(5,414))
\(\chi_{414}(5,\cdot)\)
\(\chi_{414}(11,\cdot)\)
\(\chi_{414}(65,\cdot)\)
\(\chi_{414}(83,\cdot)\)
\(\chi_{414}(113,\cdot)\)
\(\chi_{414}(149,\cdot)\)
\(\chi_{414}(155,\cdot)\)
\(\chi_{414}(191,\cdot)\)
\(\chi_{414}(203,\cdot)\)
\(\chi_{414}(221,\cdot)\)
\(\chi_{414}(227,\cdot)\)
\(\chi_{414}(245,\cdot)\)
\(\chi_{414}(263,\cdot)\)
\(\chi_{414}(281,\cdot)\)
\(\chi_{414}(293,\cdot)\)
\(\chi_{414}(329,\cdot)\)
\(\chi_{414}(365,\cdot)\)
\(\chi_{414}(383,\cdot)\)
\(\chi_{414}(389,\cdot)\)
\(\chi_{414}(401,\cdot)\)
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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 ]
\((47,235)\) → \((e\left(\frac{5}{6}\right),e\left(\frac{1}{22}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 414 }(5, a) \) |
\(1\) | \(1\) | \(e\left(\frac{7}{33}\right)\) | \(e\left(\frac{13}{66}\right)\) | \(e\left(\frac{8}{33}\right)\) | \(e\left(\frac{10}{33}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{14}{33}\right)\) | \(e\left(\frac{43}{66}\right)\) | \(e\left(\frac{31}{33}\right)\) | \(e\left(\frac{9}{22}\right)\) |
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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)
