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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(845, base_ring=CyclotomicField(52))
M = H._module
chi = DirichletCharacter(H, M([0,25]))
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pari:[g,chi] = znchar(Mod(21,845))
\(\chi_{845}(21,\cdot)\)
\(\chi_{845}(31,\cdot)\)
\(\chi_{845}(86,\cdot)\)
\(\chi_{845}(96,\cdot)\)
\(\chi_{845}(151,\cdot)\)
\(\chi_{845}(161,\cdot)\)
\(\chi_{845}(216,\cdot)\)
\(\chi_{845}(226,\cdot)\)
\(\chi_{845}(281,\cdot)\)
\(\chi_{845}(291,\cdot)\)
\(\chi_{845}(346,\cdot)\)
\(\chi_{845}(356,\cdot)\)
\(\chi_{845}(411,\cdot)\)
\(\chi_{845}(421,\cdot)\)
\(\chi_{845}(476,\cdot)\)
\(\chi_{845}(486,\cdot)\)
\(\chi_{845}(541,\cdot)\)
\(\chi_{845}(551,\cdot)\)
\(\chi_{845}(616,\cdot)\)
\(\chi_{845}(671,\cdot)\)
\(\chi_{845}(681,\cdot)\)
\(\chi_{845}(736,\cdot)\)
\(\chi_{845}(801,\cdot)\)
\(\chi_{845}(811,\cdot)\)
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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 ]
\((677,171)\) → \((1,e\left(\frac{25}{52}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(14\) |
| \( \chi_{ 845 }(21, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{25}{52}\right)\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{25}{26}\right)\) | \(e\left(\frac{5}{52}\right)\) | \(e\left(\frac{23}{52}\right)\) | \(e\left(\frac{23}{52}\right)\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{27}{52}\right)\) | \(e\left(\frac{15}{26}\right)\) | \(e\left(\frac{12}{13}\right)\) |
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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)
