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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(14157, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([0,9,11]))
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pari:[g,chi] = znchar(Mod(13771,14157))
\(\chi_{14157}(10,\cdot)\)
\(\chi_{14157}(901,\cdot)\)
\(\chi_{14157}(1297,\cdot)\)
\(\chi_{14157}(2188,\cdot)\)
\(\chi_{14157}(2584,\cdot)\)
\(\chi_{14157}(3475,\cdot)\)
\(\chi_{14157}(4762,\cdot)\)
\(\chi_{14157}(5158,\cdot)\)
\(\chi_{14157}(6445,\cdot)\)
\(\chi_{14157}(7336,\cdot)\)
\(\chi_{14157}(7732,\cdot)\)
\(\chi_{14157}(8623,\cdot)\)
\(\chi_{14157}(9019,\cdot)\)
\(\chi_{14157}(9910,\cdot)\)
\(\chi_{14157}(10306,\cdot)\)
\(\chi_{14157}(11197,\cdot)\)
\(\chi_{14157}(11593,\cdot)\)
\(\chi_{14157}(12484,\cdot)\)
\(\chi_{14157}(12880,\cdot)\)
\(\chi_{14157}(13771,\cdot)\)
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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 ]
\((6293,3511,4357)\) → \((1,e\left(\frac{3}{22}\right),e\left(\frac{1}{6}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(14\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 14157 }(13771, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{10}{33}\right)\) | \(e\left(\frac{20}{33}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{26}{33}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{59}{66}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{7}{33}\right)\) | \(e\left(\frac{1}{66}\right)\) | \(e\left(\frac{5}{33}\right)\) |
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sage:chi.jacobi_sum(n)
