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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(735, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([0,0,23]))
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pari:[g,chi] = znchar(Mod(481,735))
\(\chi_{735}(61,\cdot)\)
\(\chi_{735}(136,\cdot)\)
\(\chi_{735}(241,\cdot)\)
\(\chi_{735}(271,\cdot)\)
\(\chi_{735}(346,\cdot)\)
\(\chi_{735}(376,\cdot)\)
\(\chi_{735}(451,\cdot)\)
\(\chi_{735}(481,\cdot)\)
\(\chi_{735}(556,\cdot)\)
\(\chi_{735}(586,\cdot)\)
\(\chi_{735}(661,\cdot)\)
\(\chi_{735}(691,\cdot)\)
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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 ]
\((491,442,346)\) → \((1,1,e\left(\frac{23}{42}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(8\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) | \(22\) | \(23\) |
| \( \chi_{ 735 }(481, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{17}{21}\right)\) |
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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)
