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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8450, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([51,35]))
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pari:[g,chi] = znchar(Mod(5497,8450))
\(\chi_{8450}(427,\cdot)\)
\(\chi_{8450}(1033,\cdot)\)
\(\chi_{8450}(1263,\cdot)\)
\(\chi_{8450}(2117,\cdot)\)
\(\chi_{8450}(2347,\cdot)\)
\(\chi_{8450}(2723,\cdot)\)
\(\chi_{8450}(2953,\cdot)\)
\(\chi_{8450}(4037,\cdot)\)
\(\chi_{8450}(4413,\cdot)\)
\(\chi_{8450}(5497,\cdot)\)
\(\chi_{8450}(5727,\cdot)\)
\(\chi_{8450}(6103,\cdot)\)
\(\chi_{8450}(6333,\cdot)\)
\(\chi_{8450}(7187,\cdot)\)
\(\chi_{8450}(7417,\cdot)\)
\(\chi_{8450}(8023,\cdot)\)
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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 ]
\((677,3551)\) → \((e\left(\frac{17}{20}\right),e\left(\frac{7}{12}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 8450 }(5497, a) \) |
\(1\) | \(1\) | \(e\left(\frac{17}{60}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{41}{60}\right)\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{11}{60}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{1}{30}\right)\) |
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sage:chi.jacobi_sum(n)
