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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9386, base_ring=CyclotomicField(342))
M = H._module
chi = DirichletCharacter(H, M([285,230]))
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pari:[g,chi] = znchar(Mod(2077,9386))
\(\chi_{9386}(17,\cdot)\)
\(\chi_{9386}(23,\cdot)\)
\(\chi_{9386}(43,\cdot)\)
\(\chi_{9386}(101,\cdot)\)
\(\chi_{9386}(225,\cdot)\)
\(\chi_{9386}(465,\cdot)\)
\(\chi_{9386}(511,\cdot)\)
\(\chi_{9386}(517,\cdot)\)
\(\chi_{9386}(537,\cdot)\)
\(\chi_{9386}(719,\cdot)\)
\(\chi_{9386}(959,\cdot)\)
\(\chi_{9386}(1005,\cdot)\)
\(\chi_{9386}(1011,\cdot)\)
\(\chi_{9386}(1031,\cdot)\)
\(\chi_{9386}(1089,\cdot)\)
\(\chi_{9386}(1213,\cdot)\)
\(\chi_{9386}(1453,\cdot)\)
\(\chi_{9386}(1499,\cdot)\)
\(\chi_{9386}(1505,\cdot)\)
\(\chi_{9386}(1525,\cdot)\)
\(\chi_{9386}(1583,\cdot)\)
\(\chi_{9386}(1707,\cdot)\)
\(\chi_{9386}(1947,\cdot)\)
\(\chi_{9386}(1993,\cdot)\)
\(\chi_{9386}(1999,\cdot)\)
\(\chi_{9386}(2019,\cdot)\)
\(\chi_{9386}(2077,\cdot)\)
\(\chi_{9386}(2201,\cdot)\)
\(\chi_{9386}(2441,\cdot)\)
\(\chi_{9386}(2487,\cdot)\)
...
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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 ]
\((1445,3251)\) → \((e\left(\frac{5}{6}\right),e\left(\frac{115}{171}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(15\) | \(17\) | \(21\) | \(23\) | \(25\) |
| \( \chi_{ 9386 }(2077, a) \) |
\(1\) | \(1\) | \(e\left(\frac{139}{171}\right)\) | \(e\left(\frac{179}{342}\right)\) | \(e\left(\frac{5}{114}\right)\) | \(e\left(\frac{107}{171}\right)\) | \(e\left(\frac{49}{114}\right)\) | \(e\left(\frac{115}{342}\right)\) | \(e\left(\frac{112}{171}\right)\) | \(e\left(\frac{293}{342}\right)\) | \(e\left(\frac{89}{171}\right)\) | \(e\left(\frac{8}{171}\right)\) |
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sage:chi.jacobi_sum(n)
