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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2312, base_ring=CyclotomicField(16))
M = H._module
chi = DirichletCharacter(H, M([0,8,3]))
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pari:[g,chi] = znchar(Mod(2237,2312))
\(\chi_{2312}(653,\cdot)\)
\(\chi_{2312}(709,\cdot)\)
\(\chi_{2312}(1221,\cdot)\)
\(\chi_{2312}(1405,\cdot)\)
\(\chi_{2312}(1485,\cdot)\)
\(\chi_{2312}(1669,\cdot)\)
\(\chi_{2312}(2181,\cdot)\)
\(\chi_{2312}(2237,\cdot)\)
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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 ]
\((1735,1157,1737)\) → \((1,-1,e\left(\frac{3}{16}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(19\) | \(21\) | \(23\) |
| \( \chi_{ 2312 }(2237, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(i\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(-i\) | \(e\left(\frac{13}{16}\right)\) |
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sage:chi.jacobi_sum(n)
