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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8732, base_ring=CyclotomicField(58))
M = H._module
chi = DirichletCharacter(H, M([0,29,10]))
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pari:[g,chi] = znchar(Mod(2145,8732))
\(\chi_{8732}(369,\cdot)\)
\(\chi_{8732}(517,\cdot)\)
\(\chi_{8732}(665,\cdot)\)
\(\chi_{8732}(813,\cdot)\)
\(\chi_{8732}(961,\cdot)\)
\(\chi_{8732}(1405,\cdot)\)
\(\chi_{8732}(1553,\cdot)\)
\(\chi_{8732}(1701,\cdot)\)
\(\chi_{8732}(1849,\cdot)\)
\(\chi_{8732}(2145,\cdot)\)
\(\chi_{8732}(2293,\cdot)\)
\(\chi_{8732}(2441,\cdot)\)
\(\chi_{8732}(2885,\cdot)\)
\(\chi_{8732}(3329,\cdot)\)
\(\chi_{8732}(3625,\cdot)\)
\(\chi_{8732}(3921,\cdot)\)
\(\chi_{8732}(4069,\cdot)\)
\(\chi_{8732}(4217,\cdot)\)
\(\chi_{8732}(4513,\cdot)\)
\(\chi_{8732}(5549,\cdot)\)
\(\chi_{8732}(5845,\cdot)\)
\(\chi_{8732}(6141,\cdot)\)
\(\chi_{8732}(6289,\cdot)\)
\(\chi_{8732}(6585,\cdot)\)
\(\chi_{8732}(6733,\cdot)\)
\(\chi_{8732}(7325,\cdot)\)
\(\chi_{8732}(8065,\cdot)\)
\(\chi_{8732}(8213,\cdot)\)
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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 ]
\((4367,1889,297)\) → \((1,-1,e\left(\frac{5}{29}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 8732 }(2145, a) \) |
\(1\) | \(1\) | \(e\left(\frac{18}{29}\right)\) | \(e\left(\frac{31}{58}\right)\) | \(e\left(\frac{3}{29}\right)\) | \(e\left(\frac{7}{29}\right)\) | \(e\left(\frac{9}{29}\right)\) | \(e\left(\frac{15}{58}\right)\) | \(e\left(\frac{9}{58}\right)\) | \(e\left(\frac{23}{58}\right)\) | \(e\left(\frac{3}{58}\right)\) | \(e\left(\frac{21}{29}\right)\) |
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sage:chi.jacobi_sum(n)
