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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7803, base_ring=CyclotomicField(68))
M = H._module
chi = DirichletCharacter(H, M([0,29]))
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pari:[g,chi] = znchar(Mod(5725,7803))
\(\chi_{7803}(55,\cdot)\)
\(\chi_{7803}(217,\cdot)\)
\(\chi_{7803}(514,\cdot)\)
\(\chi_{7803}(676,\cdot)\)
\(\chi_{7803}(973,\cdot)\)
\(\chi_{7803}(1135,\cdot)\)
\(\chi_{7803}(1432,\cdot)\)
\(\chi_{7803}(1594,\cdot)\)
\(\chi_{7803}(1891,\cdot)\)
\(\chi_{7803}(2053,\cdot)\)
\(\chi_{7803}(2512,\cdot)\)
\(\chi_{7803}(2809,\cdot)\)
\(\chi_{7803}(2971,\cdot)\)
\(\chi_{7803}(3268,\cdot)\)
\(\chi_{7803}(3727,\cdot)\)
\(\chi_{7803}(3889,\cdot)\)
\(\chi_{7803}(4186,\cdot)\)
\(\chi_{7803}(4348,\cdot)\)
\(\chi_{7803}(4645,\cdot)\)
\(\chi_{7803}(4807,\cdot)\)
\(\chi_{7803}(5104,\cdot)\)
\(\chi_{7803}(5266,\cdot)\)
\(\chi_{7803}(5563,\cdot)\)
\(\chi_{7803}(5725,\cdot)\)
\(\chi_{7803}(6022,\cdot)\)
\(\chi_{7803}(6184,\cdot)\)
\(\chi_{7803}(6481,\cdot)\)
\(\chi_{7803}(6643,\cdot)\)
\(\chi_{7803}(6940,\cdot)\)
\(\chi_{7803}(7102,\cdot)\)
...
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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 ]
\((2891,2026)\) → \((1,e\left(\frac{29}{68}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 7803 }(5725, a) \) |
\(1\) | \(1\) | \(e\left(\frac{1}{34}\right)\) | \(e\left(\frac{1}{17}\right)\) | \(e\left(\frac{45}{68}\right)\) | \(e\left(\frac{7}{68}\right)\) | \(e\left(\frac{3}{34}\right)\) | \(e\left(\frac{47}{68}\right)\) | \(e\left(\frac{55}{68}\right)\) | \(e\left(\frac{10}{17}\right)\) | \(e\left(\frac{9}{68}\right)\) | \(e\left(\frac{2}{17}\right)\) |
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sage:chi.jacobi_sum(n)
