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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(230, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([33,30]))
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pari:[g,chi] = znchar(Mod(203,230))
\(\chi_{230}(7,\cdot)\)
\(\chi_{230}(17,\cdot)\)
\(\chi_{230}(33,\cdot)\)
\(\chi_{230}(37,\cdot)\)
\(\chi_{230}(43,\cdot)\)
\(\chi_{230}(53,\cdot)\)
\(\chi_{230}(57,\cdot)\)
\(\chi_{230}(63,\cdot)\)
\(\chi_{230}(67,\cdot)\)
\(\chi_{230}(83,\cdot)\)
\(\chi_{230}(97,\cdot)\)
\(\chi_{230}(103,\cdot)\)
\(\chi_{230}(107,\cdot)\)
\(\chi_{230}(113,\cdot)\)
\(\chi_{230}(143,\cdot)\)
\(\chi_{230}(153,\cdot)\)
\(\chi_{230}(157,\cdot)\)
\(\chi_{230}(203,\cdot)\)
\(\chi_{230}(217,\cdot)\)
\(\chi_{230}(227,\cdot)\)
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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 ]
\((47,51)\) → \((-i,e\left(\frac{15}{22}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(27\) | \(29\) |
| \( \chi_{ 230 }(203, a) \) |
\(1\) | \(1\) | \(e\left(\frac{7}{44}\right)\) | \(e\left(\frac{31}{44}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{35}{44}\right)\) | \(e\left(\frac{23}{44}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{21}{44}\right)\) | \(e\left(\frac{17}{22}\right)\) |
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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)
