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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7865, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([45,18,10]))
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pari:[g,chi] = znchar(Mod(3748,7865))
\(\chi_{7865}(602,\cdot)\)
\(\chi_{7865}(1492,\cdot)\)
\(\chi_{7865}(2097,\cdot)\)
\(\chi_{7865}(2272,\cdot)\)
\(\chi_{7865}(2532,\cdot)\)
\(\chi_{7865}(2877,\cdot)\)
\(\chi_{7865}(3137,\cdot)\)
\(\chi_{7865}(3143,\cdot)\)
\(\chi_{7865}(3748,\cdot)\)
\(\chi_{7865}(4638,\cdot)\)
\(\chi_{7865}(5243,\cdot)\)
\(\chi_{7865}(5418,\cdot)\)
\(\chi_{7865}(5678,\cdot)\)
\(\chi_{7865}(6023,\cdot)\)
\(\chi_{7865}(6283,\cdot)\)
\(\chi_{7865}(7862,\cdot)\)
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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 ]
\((3147,3511,1211)\) → \((-i,e\left(\frac{3}{10}\right),e\left(\frac{1}{6}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(12\) | \(14\) | \(16\) |
| \( \chi_{ 7865 }(3748, a) \) |
\(1\) | \(1\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{19}{60}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{41}{60}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(-i\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{13}{15}\right)\) |
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sage:chi.jacobi_sum(n)
