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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1309, base_ring=CyclotomicField(40))
M = H._module
chi = DirichletCharacter(H, M([0,12,25]))
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pari:[g,chi] = znchar(Mod(8,1309))
\(\chi_{1309}(8,\cdot)\)
\(\chi_{1309}(127,\cdot)\)
\(\chi_{1309}(134,\cdot)\)
\(\chi_{1309}(162,\cdot)\)
\(\chi_{1309}(281,\cdot)\)
\(\chi_{1309}(365,\cdot)\)
\(\chi_{1309}(393,\cdot)\)
\(\chi_{1309}(491,\cdot)\)
\(\chi_{1309}(512,\cdot)\)
\(\chi_{1309}(519,\cdot)\)
\(\chi_{1309}(722,\cdot)\)
\(\chi_{1309}(750,\cdot)\)
\(\chi_{1309}(876,\cdot)\)
\(\chi_{1309}(1086,\cdot)\)
\(\chi_{1309}(1107,\cdot)\)
\(\chi_{1309}(1205,\cdot)\)
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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 ]
\((1123,596,309)\) → \((1,e\left(\frac{3}{10}\right),e\left(\frac{5}{8}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(12\) | \(13\) |
| \( \chi_{ 1309 }(8, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{1}{40}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{13}{40}\right)\) | \(e\left(\frac{3}{40}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{4}{5}\right)\) |
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sage:chi.jacobi_sum(n)
