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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(14157, base_ring=CyclotomicField(110))
M = H._module
chi = DirichletCharacter(H, M([0,1,0]))
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pari:[g,chi] = znchar(Mod(3511,14157))
\(\chi_{14157}(469,\cdot)\)
\(\chi_{14157}(937,\cdot)\)
\(\chi_{14157}(1405,\cdot)\)
\(\chi_{14157}(1756,\cdot)\)
\(\chi_{14157}(2107,\cdot)\)
\(\chi_{14157}(2224,\cdot)\)
\(\chi_{14157}(2692,\cdot)\)
\(\chi_{14157}(3043,\cdot)\)
\(\chi_{14157}(3394,\cdot)\)
\(\chi_{14157}(3511,\cdot)\)
\(\chi_{14157}(3979,\cdot)\)
\(\chi_{14157}(4330,\cdot)\)
\(\chi_{14157}(4681,\cdot)\)
\(\chi_{14157}(4798,\cdot)\)
\(\chi_{14157}(5266,\cdot)\)
\(\chi_{14157}(5617,\cdot)\)
\(\chi_{14157}(5968,\cdot)\)
\(\chi_{14157}(6085,\cdot)\)
\(\chi_{14157}(6553,\cdot)\)
\(\chi_{14157}(6904,\cdot)\)
\(\chi_{14157}(7255,\cdot)\)
\(\chi_{14157}(7840,\cdot)\)
\(\chi_{14157}(8191,\cdot)\)
\(\chi_{14157}(8542,\cdot)\)
\(\chi_{14157}(8659,\cdot)\)
\(\chi_{14157}(9127,\cdot)\)
\(\chi_{14157}(9829,\cdot)\)
\(\chi_{14157}(9946,\cdot)\)
\(\chi_{14157}(10414,\cdot)\)
\(\chi_{14157}(10765,\cdot)\)
...
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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 ]
\((6293,3511,4357)\) → \((1,e\left(\frac{1}{110}\right),1)\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(14\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 14157 }(3511, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{1}{110}\right)\) | \(e\left(\frac{1}{55}\right)\) | \(e\left(\frac{37}{55}\right)\) | \(e\left(\frac{7}{110}\right)\) | \(e\left(\frac{3}{110}\right)\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{4}{55}\right)\) | \(e\left(\frac{2}{55}\right)\) | \(e\left(\frac{49}{110}\right)\) | \(e\left(\frac{83}{110}\right)\) |
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sage:chi.jacobi_sum(n)
