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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(44100, base_ring=CyclotomicField(210))
M = H._module
chi = DirichletCharacter(H, M([0,35,189,155]))
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pari:[g,chi] = znchar(Mod(8669,44100))
\(\chi_{44100}(1769,\cdot)\)
\(\chi_{44100}(2369,\cdot)\)
\(\chi_{44100}(3029,\cdot)\)
\(\chi_{44100}(3629,\cdot)\)
\(\chi_{44100}(4289,\cdot)\)
\(\chi_{44100}(4889,\cdot)\)
\(\chi_{44100}(6809,\cdot)\)
\(\chi_{44100}(7409,\cdot)\)
\(\chi_{44100}(8069,\cdot)\)
\(\chi_{44100}(8669,\cdot)\)
\(\chi_{44100}(10589,\cdot)\)
\(\chi_{44100}(11189,\cdot)\)
\(\chi_{44100}(13109,\cdot)\)
\(\chi_{44100}(13709,\cdot)\)
\(\chi_{44100}(14369,\cdot)\)
\(\chi_{44100}(14969,\cdot)\)
\(\chi_{44100}(15629,\cdot)\)
\(\chi_{44100}(16229,\cdot)\)
\(\chi_{44100}(16889,\cdot)\)
\(\chi_{44100}(17489,\cdot)\)
\(\chi_{44100}(19409,\cdot)\)
\(\chi_{44100}(20009,\cdot)\)
\(\chi_{44100}(20669,\cdot)\)
\(\chi_{44100}(21269,\cdot)\)
\(\chi_{44100}(21929,\cdot)\)
\(\chi_{44100}(22529,\cdot)\)
\(\chi_{44100}(23189,\cdot)\)
\(\chi_{44100}(23789,\cdot)\)
\(\chi_{44100}(25709,\cdot)\)
\(\chi_{44100}(26309,\cdot)\)
...
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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 ]
\((22051,34301,15877,9901)\) → \((1,e\left(\frac{1}{6}\right),e\left(\frac{9}{10}\right),e\left(\frac{31}{42}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 44100 }(8669, a) \) |
\(1\) | \(1\) | \(e\left(\frac{19}{210}\right)\) | \(e\left(\frac{83}{105}\right)\) | \(e\left(\frac{137}{210}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{82}{105}\right)\) | \(e\left(\frac{53}{210}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{151}{210}\right)\) | \(e\left(\frac{53}{105}\right)\) | \(e\left(\frac{25}{42}\right)\) |
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sage:chi.jacobi_sum(n)
