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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(216000, base_ring=CyclotomicField(600))
M = H._module
chi = DirichletCharacter(H, M([0,75,100,396]))
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pari:[g,chi] = znchar(Mod(89,216000))
\(\chi_{216000}(89,\cdot)\)
\(\chi_{216000}(1529,\cdot)\)
\(\chi_{216000}(3689,\cdot)\)
\(\chi_{216000}(4409,\cdot)\)
\(\chi_{216000}(6569,\cdot)\)
\(\chi_{216000}(8009,\cdot)\)
\(\chi_{216000}(8729,\cdot)\)
\(\chi_{216000}(10169,\cdot)\)
\(\chi_{216000}(10889,\cdot)\)
\(\chi_{216000}(12329,\cdot)\)
\(\chi_{216000}(14489,\cdot)\)
\(\chi_{216000}(15209,\cdot)\)
\(\chi_{216000}(17369,\cdot)\)
\(\chi_{216000}(18809,\cdot)\)
\(\chi_{216000}(19529,\cdot)\)
\(\chi_{216000}(20969,\cdot)\)
\(\chi_{216000}(21689,\cdot)\)
\(\chi_{216000}(23129,\cdot)\)
\(\chi_{216000}(25289,\cdot)\)
\(\chi_{216000}(26009,\cdot)\)
\(\chi_{216000}(28169,\cdot)\)
\(\chi_{216000}(29609,\cdot)\)
\(\chi_{216000}(30329,\cdot)\)
\(\chi_{216000}(31769,\cdot)\)
\(\chi_{216000}(32489,\cdot)\)
\(\chi_{216000}(33929,\cdot)\)
\(\chi_{216000}(36089,\cdot)\)
\(\chi_{216000}(36809,\cdot)\)
\(\chi_{216000}(38969,\cdot)\)
\(\chi_{216000}(40409,\cdot)\)
...
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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 ]
\((114751,202501,136001,29377)\) → \((1,e\left(\frac{1}{8}\right),e\left(\frac{1}{6}\right),e\left(\frac{33}{50}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 216000 }(89, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{1}{60}\right)\) | \(e\left(\frac{571}{600}\right)\) | \(e\left(\frac{569}{600}\right)\) | \(e\left(\frac{9}{50}\right)\) | \(e\left(\frac{151}{200}\right)\) | \(e\left(\frac{13}{300}\right)\) | \(e\left(\frac{277}{600}\right)\) | \(e\left(\frac{1}{75}\right)\) | \(e\left(\frac{53}{200}\right)\) | \(e\left(\frac{187}{300}\right)\) |
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sage:chi.jacobi_sum(n)
