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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5472, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([0,27,30,2]))
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pari:[g,chi] = znchar(Mod(1769,5472))
\(\chi_{5472}(41,\cdot)\)
\(\chi_{5472}(185,\cdot)\)
\(\chi_{5472}(281,\cdot)\)
\(\chi_{5472}(713,\cdot)\)
\(\chi_{5472}(1001,\cdot)\)
\(\chi_{5472}(1769,\cdot)\)
\(\chi_{5472}(2777,\cdot)\)
\(\chi_{5472}(2921,\cdot)\)
\(\chi_{5472}(3017,\cdot)\)
\(\chi_{5472}(3449,\cdot)\)
\(\chi_{5472}(3737,\cdot)\)
\(\chi_{5472}(4505,\cdot)\)
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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 ]
\((4447,2053,1217,3745)\) → \((1,-i,e\left(\frac{5}{6}\right),e\left(\frac{1}{18}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 5472 }(1769, a) \) |
\(1\) | \(1\) | \(e\left(\frac{29}{36}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(i\) | \(e\left(\frac{7}{36}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{1}{36}\right)\) | \(-1\) | \(e\left(\frac{35}{36}\right)\) |
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sage:chi.jacobi_sum(n)
