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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(363, base_ring=CyclotomicField(110))
M = H._module
chi = DirichletCharacter(H, M([0,109]))
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pari:[g,chi] = znchar(Mod(61,363))
\(\chi_{363}(7,\cdot)\)
\(\chi_{363}(13,\cdot)\)
\(\chi_{363}(19,\cdot)\)
\(\chi_{363}(28,\cdot)\)
\(\chi_{363}(46,\cdot)\)
\(\chi_{363}(52,\cdot)\)
\(\chi_{363}(61,\cdot)\)
\(\chi_{363}(73,\cdot)\)
\(\chi_{363}(79,\cdot)\)
\(\chi_{363}(85,\cdot)\)
\(\chi_{363}(106,\cdot)\)
\(\chi_{363}(127,\cdot)\)
\(\chi_{363}(139,\cdot)\)
\(\chi_{363}(145,\cdot)\)
\(\chi_{363}(151,\cdot)\)
\(\chi_{363}(160,\cdot)\)
\(\chi_{363}(172,\cdot)\)
\(\chi_{363}(178,\cdot)\)
\(\chi_{363}(184,\cdot)\)
\(\chi_{363}(193,\cdot)\)
\(\chi_{363}(205,\cdot)\)
\(\chi_{363}(211,\cdot)\)
\(\chi_{363}(217,\cdot)\)
\(\chi_{363}(226,\cdot)\)
\(\chi_{363}(238,\cdot)\)
\(\chi_{363}(244,\cdot)\)
\(\chi_{363}(250,\cdot)\)
\(\chi_{363}(259,\cdot)\)
\(\chi_{363}(271,\cdot)\)
\(\chi_{363}(277,\cdot)\)
...
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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 ]
\((122,244)\) → \((1,e\left(\frac{109}{110}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(13\) | \(14\) | \(16\) | \(17\) |
| \( \chi_{ 363 }(61, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{109}{110}\right)\) | \(e\left(\frac{54}{55}\right)\) | \(e\left(\frac{18}{55}\right)\) | \(e\left(\frac{103}{110}\right)\) | \(e\left(\frac{107}{110}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{9}{110}\right)\) | \(e\left(\frac{51}{55}\right)\) | \(e\left(\frac{53}{55}\right)\) | \(e\left(\frac{61}{110}\right)\) |
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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)
