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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8112, base_ring=CyclotomicField(78))
M = H._module
chi = DirichletCharacter(H, M([0,0,0,43]))
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pari:[g,chi] = znchar(Mod(6673,8112))
\(\chi_{8112}(49,\cdot)\)
\(\chi_{8112}(433,\cdot)\)
\(\chi_{8112}(673,\cdot)\)
\(\chi_{8112}(1057,\cdot)\)
\(\chi_{8112}(1297,\cdot)\)
\(\chi_{8112}(1681,\cdot)\)
\(\chi_{8112}(1921,\cdot)\)
\(\chi_{8112}(2305,\cdot)\)
\(\chi_{8112}(2545,\cdot)\)
\(\chi_{8112}(2929,\cdot)\)
\(\chi_{8112}(3169,\cdot)\)
\(\chi_{8112}(3553,\cdot)\)
\(\chi_{8112}(3793,\cdot)\)
\(\chi_{8112}(4177,\cdot)\)
\(\chi_{8112}(4801,\cdot)\)
\(\chi_{8112}(5041,\cdot)\)
\(\chi_{8112}(5425,\cdot)\)
\(\chi_{8112}(5665,\cdot)\)
\(\chi_{8112}(6049,\cdot)\)
\(\chi_{8112}(6289,\cdot)\)
\(\chi_{8112}(6673,\cdot)\)
\(\chi_{8112}(6913,\cdot)\)
\(\chi_{8112}(7297,\cdot)\)
\(\chi_{8112}(7537,\cdot)\)
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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 ]
\((5071,6085,2705,3889)\) → \((1,1,1,e\left(\frac{43}{78}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 8112 }(6673, a) \) |
\(1\) | \(1\) | \(e\left(\frac{25}{26}\right)\) | \(e\left(\frac{77}{78}\right)\) | \(e\left(\frac{61}{78}\right)\) | \(e\left(\frac{19}{39}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{2}{39}\right)\) | \(e\left(\frac{15}{26}\right)\) | \(e\left(\frac{37}{39}\right)\) |
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sage:chi.jacobi_sum(n)
