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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([16,0,45]))
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pari:[g,chi] = znchar(Mod(23,1309))
\(\chi_{1309}(23,\cdot)\)
\(\chi_{1309}(177,\cdot)\)
\(\chi_{1309}(452,\cdot)\)
\(\chi_{1309}(606,\cdot)\)
\(\chi_{1309}(639,\cdot)\)
\(\chi_{1309}(683,\cdot)\)
\(\chi_{1309}(760,\cdot)\)
\(\chi_{1309}(793,\cdot)\)
\(\chi_{1309}(870,\cdot)\)
\(\chi_{1309}(947,\cdot)\)
\(\chi_{1309}(991,\cdot)\)
\(\chi_{1309}(1068,\cdot)\)
\(\chi_{1309}(1145,\cdot)\)
\(\chi_{1309}(1178,\cdot)\)
\(\chi_{1309}(1255,\cdot)\)
\(\chi_{1309}(1299,\cdot)\)
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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 ]
\((1123,596,309)\) → \((e\left(\frac{1}{3}\right),1,e\left(\frac{15}{16}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(12\) | \(13\) |
| \( \chi_{ 1309 }(23, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{19}{24}\right)\) | \(e\left(\frac{13}{48}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{17}{48}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{7}{48}\right)\) | \(e\left(\frac{41}{48}\right)\) | \(-i\) |
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sage:chi.jacobi_sum(n)
