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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1734, base_ring=CyclotomicField(272))
M = H._module
chi = DirichletCharacter(H, M([0,251]))
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pari:[g,chi] = znchar(Mod(211,1734))
\(\chi_{1734}(7,\cdot)\)
\(\chi_{1734}(31,\cdot)\)
\(\chi_{1734}(37,\cdot)\)
\(\chi_{1734}(61,\cdot)\)
\(\chi_{1734}(73,\cdot)\)
\(\chi_{1734}(79,\cdot)\)
\(\chi_{1734}(91,\cdot)\)
\(\chi_{1734}(97,\cdot)\)
\(\chi_{1734}(109,\cdot)\)
\(\chi_{1734}(133,\cdot)\)
\(\chi_{1734}(139,\cdot)\)
\(\chi_{1734}(163,\cdot)\)
\(\chi_{1734}(175,\cdot)\)
\(\chi_{1734}(181,\cdot)\)
\(\chi_{1734}(193,\cdot)\)
\(\chi_{1734}(199,\cdot)\)
\(\chi_{1734}(211,\cdot)\)
\(\chi_{1734}(235,\cdot)\)
\(\chi_{1734}(241,\cdot)\)
\(\chi_{1734}(265,\cdot)\)
\(\chi_{1734}(277,\cdot)\)
\(\chi_{1734}(283,\cdot)\)
\(\chi_{1734}(295,\cdot)\)
\(\chi_{1734}(301,\cdot)\)
\(\chi_{1734}(313,\cdot)\)
\(\chi_{1734}(337,\cdot)\)
\(\chi_{1734}(343,\cdot)\)
\(\chi_{1734}(367,\cdot)\)
\(\chi_{1734}(379,\cdot)\)
\(\chi_{1734}(385,\cdot)\)
...
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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 ]
\((1157,1159)\) → \((1,e\left(\frac{251}{272}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 1734 }(211, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{87}{272}\right)\) | \(e\left(\frac{9}{272}\right)\) | \(e\left(\frac{61}{272}\right)\) | \(e\left(\frac{59}{68}\right)\) | \(e\left(\frac{125}{136}\right)\) | \(e\left(\frac{213}{272}\right)\) | \(e\left(\frac{87}{136}\right)\) | \(e\left(\frac{95}{272}\right)\) | \(e\left(\frac{83}{272}\right)\) | \(e\left(\frac{6}{17}\right)\) |
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sage:chi.jacobi_sum(n)
