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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4225, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([57,25]))
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pari:[g,chi] = znchar(Mod(188,4225))
\(\chi_{4225}(188,\cdot)\)
\(\chi_{4225}(427,\cdot)\)
\(\chi_{4225}(1033,\cdot)\)
\(\chi_{4225}(1263,\cdot)\)
\(\chi_{4225}(1272,\cdot)\)
\(\chi_{4225}(1502,\cdot)\)
\(\chi_{4225}(1878,\cdot)\)
\(\chi_{4225}(2108,\cdot)\)
\(\chi_{4225}(2117,\cdot)\)
\(\chi_{4225}(2347,\cdot)\)
\(\chi_{4225}(2723,\cdot)\)
\(\chi_{4225}(2953,\cdot)\)
\(\chi_{4225}(2962,\cdot)\)
\(\chi_{4225}(3192,\cdot)\)
\(\chi_{4225}(3798,\cdot)\)
\(\chi_{4225}(4037,\cdot)\)
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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 ]
\((677,3551)\) → \((e\left(\frac{19}{20}\right),e\left(\frac{5}{12}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(14\) |
| \( \chi_{ 4225 }(188, a) \) |
\(1\) | \(1\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{19}{60}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{41}{60}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{7}{60}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{7}{10}\right)\) |
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sage:chi.jacobi_sum(n)
