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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(270, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([28,27]))
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pari:[g,chi] = znchar(Mod(103,270))
\(\chi_{270}(7,\cdot)\)
\(\chi_{270}(13,\cdot)\)
\(\chi_{270}(43,\cdot)\)
\(\chi_{270}(67,\cdot)\)
\(\chi_{270}(97,\cdot)\)
\(\chi_{270}(103,\cdot)\)
\(\chi_{270}(133,\cdot)\)
\(\chi_{270}(157,\cdot)\)
\(\chi_{270}(187,\cdot)\)
\(\chi_{270}(193,\cdot)\)
\(\chi_{270}(223,\cdot)\)
\(\chi_{270}(247,\cdot)\)
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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 ]
\((191,217)\) → \((e\left(\frac{7}{9}\right),-i)\)
| \(a\) |
\(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 270 }(103, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{7}{36}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{17}{36}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{29}{36}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{2}{9}\right)\) |
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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)
