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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2160, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([18,18,22,27]))
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pari:[g,chi] = znchar(Mod(23,2160))
\(\chi_{2160}(23,\cdot)\)
\(\chi_{2160}(167,\cdot)\)
\(\chi_{2160}(263,\cdot)\)
\(\chi_{2160}(407,\cdot)\)
\(\chi_{2160}(743,\cdot)\)
\(\chi_{2160}(887,\cdot)\)
\(\chi_{2160}(983,\cdot)\)
\(\chi_{2160}(1127,\cdot)\)
\(\chi_{2160}(1463,\cdot)\)
\(\chi_{2160}(1607,\cdot)\)
\(\chi_{2160}(1703,\cdot)\)
\(\chi_{2160}(1847,\cdot)\)
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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 ]
\((271,1621,2081,1297)\) → \((-1,-1,e\left(\frac{11}{18}\right),-i)\)
| \(a\) |
\(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 2160 }(23, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{1}{36}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{23}{36}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{17}{36}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{7}{18}\right)\) |
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sage:chi.jacobi_sum(n)
