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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(834, base_ring=CyclotomicField(138))
M = H._module
chi = DirichletCharacter(H, M([69,104]))
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pari:[g,chi] = znchar(Mod(89,834))
\(\chi_{834}(5,\cdot)\)
\(\chi_{834}(11,\cdot)\)
\(\chi_{834}(29,\cdot)\)
\(\chi_{834}(35,\cdot)\)
\(\chi_{834}(41,\cdot)\)
\(\chi_{834}(47,\cdot)\)
\(\chi_{834}(71,\cdot)\)
\(\chi_{834}(83,\cdot)\)
\(\chi_{834}(89,\cdot)\)
\(\chi_{834}(107,\cdot)\)
\(\chi_{834}(113,\cdot)\)
\(\chi_{834}(137,\cdot)\)
\(\chi_{834}(143,\cdot)\)
\(\chi_{834}(155,\cdot)\)
\(\chi_{834}(167,\cdot)\)
\(\chi_{834}(185,\cdot)\)
\(\chi_{834}(257,\cdot)\)
\(\chi_{834}(263,\cdot)\)
\(\chi_{834}(275,\cdot)\)
\(\chi_{834}(287,\cdot)\)
\(\chi_{834}(329,\cdot)\)
\(\chi_{834}(347,\cdot)\)
\(\chi_{834}(359,\cdot)\)
\(\chi_{834}(377,\cdot)\)
\(\chi_{834}(395,\cdot)\)
\(\chi_{834}(437,\cdot)\)
\(\chi_{834}(455,\cdot)\)
\(\chi_{834}(503,\cdot)\)
\(\chi_{834}(539,\cdot)\)
\(\chi_{834}(563,\cdot)\)
...
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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 ]
\((557,697)\) → \((-1,e\left(\frac{52}{69}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
| \( \chi_{ 834 }(89, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{43}{138}\right)\) | \(e\left(\frac{47}{69}\right)\) | \(e\left(\frac{107}{138}\right)\) | \(e\left(\frac{16}{69}\right)\) | \(e\left(\frac{19}{138}\right)\) | \(e\left(\frac{67}{69}\right)\) | \(e\left(\frac{39}{46}\right)\) | \(e\left(\frac{43}{69}\right)\) | \(e\left(\frac{47}{138}\right)\) | \(e\left(\frac{14}{69}\right)\) |
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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)
