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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1014, base_ring=CyclotomicField(156))
M = H._module
chi = DirichletCharacter(H, M([0,1]))
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pari:[g,chi] = znchar(Mod(847,1014))
\(\chi_{1014}(7,\cdot)\)
\(\chi_{1014}(37,\cdot)\)
\(\chi_{1014}(67,\cdot)\)
\(\chi_{1014}(85,\cdot)\)
\(\chi_{1014}(97,\cdot)\)
\(\chi_{1014}(115,\cdot)\)
\(\chi_{1014}(145,\cdot)\)
\(\chi_{1014}(163,\cdot)\)
\(\chi_{1014}(175,\cdot)\)
\(\chi_{1014}(193,\cdot)\)
\(\chi_{1014}(223,\cdot)\)
\(\chi_{1014}(241,\cdot)\)
\(\chi_{1014}(253,\cdot)\)
\(\chi_{1014}(271,\cdot)\)
\(\chi_{1014}(301,\cdot)\)
\(\chi_{1014}(331,\cdot)\)
\(\chi_{1014}(349,\cdot)\)
\(\chi_{1014}(379,\cdot)\)
\(\chi_{1014}(397,\cdot)\)
\(\chi_{1014}(409,\cdot)\)
\(\chi_{1014}(457,\cdot)\)
\(\chi_{1014}(475,\cdot)\)
\(\chi_{1014}(487,\cdot)\)
\(\chi_{1014}(505,\cdot)\)
\(\chi_{1014}(535,\cdot)\)
\(\chi_{1014}(553,\cdot)\)
\(\chi_{1014}(565,\cdot)\)
\(\chi_{1014}(583,\cdot)\)
\(\chi_{1014}(613,\cdot)\)
\(\chi_{1014}(631,\cdot)\)
...
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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 ]
\((677,847)\) → \((1,e\left(\frac{1}{156}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 1014 }(847, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{3}{52}\right)\) | \(e\left(\frac{107}{156}\right)\) | \(e\left(\frac{103}{156}\right)\) | \(e\left(\frac{73}{78}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{3}{26}\right)\) | \(e\left(\frac{10}{39}\right)\) | \(e\left(\frac{7}{52}\right)\) | \(e\left(\frac{29}{39}\right)\) |
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sage:chi.jacobi_sum(n)
