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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1650, base_ring=CyclotomicField(20))
M = H._module
chi = DirichletCharacter(H, M([10,1,8]))
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pari:[g,chi] = znchar(Mod(1127,1650))
\(\chi_{1650}(47,\cdot)\)
\(\chi_{1650}(713,\cdot)\)
\(\chi_{1650}(917,\cdot)\)
\(\chi_{1650}(983,\cdot)\)
\(\chi_{1650}(1103,\cdot)\)
\(\chi_{1650}(1127,\cdot)\)
\(\chi_{1650}(1523,\cdot)\)
\(\chi_{1650}(1637,\cdot)\)
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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 ]
\((551,727,1201)\) → \((-1,e\left(\frac{1}{20}\right),e\left(\frac{2}{5}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(7\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 1650 }(1127, a) \) |
\(1\) | \(1\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(-i\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(i\) | \(e\left(\frac{9}{10}\right)\) | \(-i\) |
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sage:chi.jacobi_sum(n)
