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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1309, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([40,0,39]))
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pari:[g,chi] = znchar(Mod(12,1309))
\(\chi_{1309}(12,\cdot)\)
\(\chi_{1309}(45,\cdot)\)
\(\chi_{1309}(122,\cdot)\)
\(\chi_{1309}(199,\cdot)\)
\(\chi_{1309}(243,\cdot)\)
\(\chi_{1309}(320,\cdot)\)
\(\chi_{1309}(397,\cdot)\)
\(\chi_{1309}(430,\cdot)\)
\(\chi_{1309}(507,\cdot)\)
\(\chi_{1309}(551,\cdot)\)
\(\chi_{1309}(584,\cdot)\)
\(\chi_{1309}(738,\cdot)\)
\(\chi_{1309}(1013,\cdot)\)
\(\chi_{1309}(1167,\cdot)\)
\(\chi_{1309}(1200,\cdot)\)
\(\chi_{1309}(1244,\cdot)\)
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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 ]
\((1123,596,309)\) → \((e\left(\frac{5}{6}\right),1,e\left(\frac{13}{16}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(12\) | \(13\) |
| \( \chi_{ 1309 }(12, a) \) |
\(1\) | \(1\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{31}{48}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{11}{48}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{7}{24}\right)\) | \(e\left(\frac{13}{48}\right)\) | \(e\left(\frac{35}{48}\right)\) | \(-i\) |
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sage:chi.jacobi_sum(n)
