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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8112, base_ring=CyclotomicField(26))
M = H._module
chi = DirichletCharacter(H, M([13,0,13,11]))
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pari:[g,chi] = znchar(Mod(5615,8112))
\(\chi_{8112}(623,\cdot)\)
\(\chi_{8112}(1247,\cdot)\)
\(\chi_{8112}(1871,\cdot)\)
\(\chi_{8112}(2495,\cdot)\)
\(\chi_{8112}(3119,\cdot)\)
\(\chi_{8112}(3743,\cdot)\)
\(\chi_{8112}(4367,\cdot)\)
\(\chi_{8112}(4991,\cdot)\)
\(\chi_{8112}(5615,\cdot)\)
\(\chi_{8112}(6239,\cdot)\)
\(\chi_{8112}(6863,\cdot)\)
\(\chi_{8112}(7487,\cdot)\)
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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 ]
\((5071,6085,2705,3889)\) → \((-1,1,-1,e\left(\frac{11}{26}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 8112 }(5615, a) \) |
\(1\) | \(1\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{10}{13}\right)\) | \(e\left(\frac{15}{26}\right)\) | \(e\left(\frac{7}{26}\right)\) | \(1\) | \(1\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{11}{26}\right)\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{1}{13}\right)\) |
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sage:chi.jacobi_sum(n)
