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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(508288, base_ring=CyclotomicField(480))
M = H._module
chi = DirichletCharacter(H, M([0,165,384,400]))
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pari:[g,chi] = znchar(Mod(31837,508288))
\(\chi_{508288}(69,\cdot)\)
\(\chi_{508288}(2957,\cdot)\)
\(\chi_{508288}(5845,\cdot)\)
\(\chi_{508288}(8957,\cdot)\)
\(\chi_{508288}(11621,\cdot)\)
\(\chi_{508288}(11845,\cdot)\)
\(\chi_{508288}(14733,\cdot)\)
\(\chi_{508288}(20509,\cdot)\)
\(\chi_{508288}(31837,\cdot)\)
\(\chi_{508288}(34725,\cdot)\)
\(\chi_{508288}(37613,\cdot)\)
\(\chi_{508288}(40725,\cdot)\)
\(\chi_{508288}(43389,\cdot)\)
\(\chi_{508288}(43613,\cdot)\)
\(\chi_{508288}(46501,\cdot)\)
\(\chi_{508288}(52277,\cdot)\)
\(\chi_{508288}(63605,\cdot)\)
\(\chi_{508288}(66493,\cdot)\)
\(\chi_{508288}(69381,\cdot)\)
\(\chi_{508288}(72493,\cdot)\)
\(\chi_{508288}(75157,\cdot)\)
\(\chi_{508288}(75381,\cdot)\)
\(\chi_{508288}(78269,\cdot)\)
\(\chi_{508288}(84045,\cdot)\)
\(\chi_{508288}(95373,\cdot)\)
\(\chi_{508288}(98261,\cdot)\)
\(\chi_{508288}(101149,\cdot)\)
\(\chi_{508288}(104261,\cdot)\)
\(\chi_{508288}(106925,\cdot)\)
\(\chi_{508288}(107149,\cdot)\)
...
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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 ]
\((166783,174725,323457,14081)\) → \((1,e\left(\frac{11}{32}\right),e\left(\frac{4}{5}\right),e\left(\frac{5}{6}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(13\) | \(15\) | \(17\) | \(21\) | \(23\) | \(25\) |
| \( \chi_{ 508288 }(31837, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{127}{480}\right)\) | \(e\left(\frac{421}{480}\right)\) | \(e\left(\frac{3}{80}\right)\) | \(e\left(\frac{127}{240}\right)\) | \(e\left(\frac{59}{480}\right)\) | \(e\left(\frac{17}{120}\right)\) | \(e\left(\frac{19}{120}\right)\) | \(e\left(\frac{29}{96}\right)\) | \(e\left(\frac{23}{48}\right)\) | \(e\left(\frac{181}{240}\right)\) |
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sage:chi.jacobi_sum(n)
