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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8450, base_ring=CyclotomicField(52))
M = H._module
chi = DirichletCharacter(H, M([39,3]))
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pari:[g,chi] = znchar(Mod(343,8450))
\(\chi_{8450}(307,\cdot)\)
\(\chi_{8450}(343,\cdot)\)
\(\chi_{8450}(957,\cdot)\)
\(\chi_{8450}(993,\cdot)\)
\(\chi_{8450}(1607,\cdot)\)
\(\chi_{8450}(1643,\cdot)\)
\(\chi_{8450}(2257,\cdot)\)
\(\chi_{8450}(2293,\cdot)\)
\(\chi_{8450}(2907,\cdot)\)
\(\chi_{8450}(3557,\cdot)\)
\(\chi_{8450}(3593,\cdot)\)
\(\chi_{8450}(4207,\cdot)\)
\(\chi_{8450}(4243,\cdot)\)
\(\chi_{8450}(4857,\cdot)\)
\(\chi_{8450}(4893,\cdot)\)
\(\chi_{8450}(5543,\cdot)\)
\(\chi_{8450}(6157,\cdot)\)
\(\chi_{8450}(6193,\cdot)\)
\(\chi_{8450}(6807,\cdot)\)
\(\chi_{8450}(6843,\cdot)\)
\(\chi_{8450}(7457,\cdot)\)
\(\chi_{8450}(7493,\cdot)\)
\(\chi_{8450}(8107,\cdot)\)
\(\chi_{8450}(8143,\cdot)\)
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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 ]
\((677,3551)\) → \((-i,e\left(\frac{3}{52}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 8450 }(343, a) \) |
\(1\) | \(1\) | \(e\left(\frac{21}{52}\right)\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{21}{26}\right)\) | \(e\left(\frac{49}{52}\right)\) | \(e\left(\frac{9}{52}\right)\) | \(i\) | \(e\left(\frac{17}{52}\right)\) | \(-i\) | \(e\left(\frac{11}{52}\right)\) | \(e\left(\frac{21}{26}\right)\) |
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sage:chi.jacobi_sum(n)
