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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(324, base_ring=CyclotomicField(54))
M = H._module
chi = DirichletCharacter(H, M([0,38]))
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pari:[g,chi] = znchar(Mod(301,324))
\(\chi_{324}(13,\cdot)\)
\(\chi_{324}(25,\cdot)\)
\(\chi_{324}(49,\cdot)\)
\(\chi_{324}(61,\cdot)\)
\(\chi_{324}(85,\cdot)\)
\(\chi_{324}(97,\cdot)\)
\(\chi_{324}(121,\cdot)\)
\(\chi_{324}(133,\cdot)\)
\(\chi_{324}(157,\cdot)\)
\(\chi_{324}(169,\cdot)\)
\(\chi_{324}(193,\cdot)\)
\(\chi_{324}(205,\cdot)\)
\(\chi_{324}(229,\cdot)\)
\(\chi_{324}(241,\cdot)\)
\(\chi_{324}(265,\cdot)\)
\(\chi_{324}(277,\cdot)\)
\(\chi_{324}(301,\cdot)\)
\(\chi_{324}(313,\cdot)\)
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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 ]
\((163,245)\) → \((1,e\left(\frac{19}{27}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
| \( \chi_{ 324 }(301, a) \) |
\(1\) | \(1\) | \(e\left(\frac{5}{27}\right)\) | \(e\left(\frac{7}{27}\right)\) | \(e\left(\frac{4}{27}\right)\) | \(e\left(\frac{17}{27}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{20}{27}\right)\) | \(e\left(\frac{10}{27}\right)\) | \(e\left(\frac{1}{27}\right)\) | \(e\left(\frac{2}{27}\right)\) |
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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)
