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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(678, base_ring=CyclotomicField(56))
M = H._module
chi = DirichletCharacter(H, M([0,51]))
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pari:[g,chi] = znchar(Mod(61,678))
\(\chi_{678}(13,\cdot)\)
\(\chi_{678}(25,\cdot)\)
\(\chi_{678}(31,\cdot)\)
\(\chi_{678}(61,\cdot)\)
\(\chi_{678}(91,\cdot)\)
\(\chi_{678}(139,\cdot)\)
\(\chi_{678}(163,\cdot)\)
\(\chi_{678}(175,\cdot)\)
\(\chi_{678}(217,\cdot)\)
\(\chi_{678}(235,\cdot)\)
\(\chi_{678}(277,\cdot)\)
\(\chi_{678}(289,\cdot)\)
\(\chi_{678}(313,\cdot)\)
\(\chi_{678}(361,\cdot)\)
\(\chi_{678}(391,\cdot)\)
\(\chi_{678}(421,\cdot)\)
\(\chi_{678}(427,\cdot)\)
\(\chi_{678}(439,\cdot)\)
\(\chi_{678}(463,\cdot)\)
\(\chi_{678}(493,\cdot)\)
\(\chi_{678}(529,\cdot)\)
\(\chi_{678}(601,\cdot)\)
\(\chi_{678}(637,\cdot)\)
\(\chi_{678}(667,\cdot)\)
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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 ]
\((227,229)\) → \((1,e\left(\frac{51}{56}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
| \( \chi_{ 678 }(61, a) \) |
\(1\) | \(1\) | \(e\left(\frac{33}{56}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{11}{28}\right)\) | \(e\left(\frac{1}{28}\right)\) | \(e\left(\frac{31}{56}\right)\) | \(e\left(\frac{9}{56}\right)\) | \(e\left(\frac{19}{56}\right)\) | \(e\left(\frac{5}{28}\right)\) | \(e\left(\frac{3}{56}\right)\) | \(e\left(\frac{15}{28}\right)\) |
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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)
