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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1682, base_ring=CyclotomicField(406))
M = H._module
chi = DirichletCharacter(H, M([402]))
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pari:[g,chi] = znchar(Mod(23,1682))
\(\chi_{1682}(7,\cdot)\)
\(\chi_{1682}(23,\cdot)\)
\(\chi_{1682}(25,\cdot)\)
\(\chi_{1682}(45,\cdot)\)
\(\chi_{1682}(49,\cdot)\)
\(\chi_{1682}(53,\cdot)\)
\(\chi_{1682}(65,\cdot)\)
\(\chi_{1682}(81,\cdot)\)
\(\chi_{1682}(83,\cdot)\)
\(\chi_{1682}(103,\cdot)\)
\(\chi_{1682}(107,\cdot)\)
\(\chi_{1682}(111,\cdot)\)
\(\chi_{1682}(123,\cdot)\)
\(\chi_{1682}(139,\cdot)\)
\(\chi_{1682}(141,\cdot)\)
\(\chi_{1682}(161,\cdot)\)
\(\chi_{1682}(165,\cdot)\)
\(\chi_{1682}(169,\cdot)\)
\(\chi_{1682}(181,\cdot)\)
\(\chi_{1682}(197,\cdot)\)
\(\chi_{1682}(199,\cdot)\)
\(\chi_{1682}(219,\cdot)\)
\(\chi_{1682}(223,\cdot)\)
\(\chi_{1682}(227,\cdot)\)
\(\chi_{1682}(239,\cdot)\)
\(\chi_{1682}(255,\cdot)\)
\(\chi_{1682}(257,\cdot)\)
\(\chi_{1682}(277,\cdot)\)
\(\chi_{1682}(281,\cdot)\)
\(\chi_{1682}(285,\cdot)\)
...
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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 ]
\(843\) → \(e\left(\frac{201}{203}\right)\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 1682 }(23, a) \) |
\(1\) | \(1\) | \(e\left(\frac{144}{203}\right)\) | \(e\left(\frac{5}{203}\right)\) | \(e\left(\frac{60}{203}\right)\) | \(e\left(\frac{85}{203}\right)\) | \(e\left(\frac{41}{203}\right)\) | \(e\left(\frac{97}{203}\right)\) | \(e\left(\frac{149}{203}\right)\) | \(e\left(\frac{11}{29}\right)\) | \(e\left(\frac{171}{203}\right)\) | \(e\left(\frac{1}{203}\right)\) |
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sage:chi.jacobi_sum(n)
