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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(15730, base_ring=CyclotomicField(110))
M = H._module
chi = DirichletCharacter(H, M([0,46,55]))
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pari:[g,chi] = znchar(Mod(11361,15730))
\(\chi_{15730}(181,\cdot)\)
\(\chi_{15730}(311,\cdot)\)
\(\chi_{15730}(961,\cdot)\)
\(\chi_{15730}(1351,\cdot)\)
\(\chi_{15730}(1611,\cdot)\)
\(\chi_{15730}(1741,\cdot)\)
\(\chi_{15730}(2391,\cdot)\)
\(\chi_{15730}(2781,\cdot)\)
\(\chi_{15730}(3041,\cdot)\)
\(\chi_{15730}(3171,\cdot)\)
\(\chi_{15730}(3821,\cdot)\)
\(\chi_{15730}(4211,\cdot)\)
\(\chi_{15730}(4471,\cdot)\)
\(\chi_{15730}(5251,\cdot)\)
\(\chi_{15730}(5641,\cdot)\)
\(\chi_{15730}(5901,\cdot)\)
\(\chi_{15730}(6031,\cdot)\)
\(\chi_{15730}(6681,\cdot)\)
\(\chi_{15730}(7071,\cdot)\)
\(\chi_{15730}(7331,\cdot)\)
\(\chi_{15730}(7461,\cdot)\)
\(\chi_{15730}(8111,\cdot)\)
\(\chi_{15730}(8501,\cdot)\)
\(\chi_{15730}(8761,\cdot)\)
\(\chi_{15730}(8891,\cdot)\)
\(\chi_{15730}(9541,\cdot)\)
\(\chi_{15730}(10321,\cdot)\)
\(\chi_{15730}(11361,\cdot)\)
\(\chi_{15730}(11621,\cdot)\)
\(\chi_{15730}(11751,\cdot)\)
...
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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 ]
\((3147,3511,1211)\) → \((1,e\left(\frac{23}{55}\right),-1)\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) | \(31\) |
| \( \chi_{ 15730 }(11361, a) \) |
\(1\) | \(1\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{47}{110}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{27}{55}\right)\) | \(e\left(\frac{23}{110}\right)\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{6}{55}\right)\) | \(e\left(\frac{51}{110}\right)\) |
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sage:chi.jacobi_sum(n)
