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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([0,1,6]))
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pari:[g,chi] = znchar(Mod(877,1650))
\(\chi_{1650}(337,\cdot)\)
\(\chi_{1650}(877,\cdot)\)
\(\chi_{1650}(1003,\cdot)\)
\(\chi_{1650}(1063,\cdot)\)
\(\chi_{1650}(1117,\cdot)\)
\(\chi_{1650}(1183,\cdot)\)
\(\chi_{1650}(1273,\cdot)\)
\(\chi_{1650}(1597,\cdot)\)
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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 ]
\((551,727,1201)\) → \((1,e\left(\frac{1}{20}\right),e\left(\frac{3}{10}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(7\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 1650 }(877, a) \) |
\(1\) | \(1\) | \(e\left(\frac{7}{20}\right)\) | \(i\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(i\) |
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sage:chi.jacobi_sum(n)
