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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(14700, base_ring=CyclotomicField(140))
M = H._module
chi = DirichletCharacter(H, M([0,0,133,110]))
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pari:[g,chi] = znchar(Mod(13,14700))
\(\chi_{14700}(13,\cdot)\)
\(\chi_{14700}(433,\cdot)\)
\(\chi_{14700}(517,\cdot)\)
\(\chi_{14700}(853,\cdot)\)
\(\chi_{14700}(937,\cdot)\)
\(\chi_{14700}(1777,\cdot)\)
\(\chi_{14700}(2113,\cdot)\)
\(\chi_{14700}(2197,\cdot)\)
\(\chi_{14700}(2533,\cdot)\)
\(\chi_{14700}(2617,\cdot)\)
\(\chi_{14700}(2953,\cdot)\)
\(\chi_{14700}(3373,\cdot)\)
\(\chi_{14700}(3877,\cdot)\)
\(\chi_{14700}(4297,\cdot)\)
\(\chi_{14700}(4633,\cdot)\)
\(\chi_{14700}(4717,\cdot)\)
\(\chi_{14700}(5053,\cdot)\)
\(\chi_{14700}(5137,\cdot)\)
\(\chi_{14700}(5473,\cdot)\)
\(\chi_{14700}(6313,\cdot)\)
\(\chi_{14700}(6397,\cdot)\)
\(\chi_{14700}(6733,\cdot)\)
\(\chi_{14700}(6817,\cdot)\)
\(\chi_{14700}(7237,\cdot)\)
\(\chi_{14700}(7573,\cdot)\)
\(\chi_{14700}(8077,\cdot)\)
\(\chi_{14700}(8413,\cdot)\)
\(\chi_{14700}(8497,\cdot)\)
\(\chi_{14700}(8833,\cdot)\)
\(\chi_{14700}(9253,\cdot)\)
...
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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 ]
\((7351,4901,1177,9901)\) → \((1,1,e\left(\frac{19}{20}\right),e\left(\frac{11}{14}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 14700 }(13, a) \) |
\(1\) | \(1\) | \(e\left(\frac{22}{35}\right)\) | \(e\left(\frac{137}{140}\right)\) | \(e\left(\frac{139}{140}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{43}{140}\right)\) | \(e\left(\frac{3}{70}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{97}{140}\right)\) | \(e\left(\frac{41}{70}\right)\) | \(e\left(\frac{27}{28}\right)\) |
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sage:chi.jacobi_sum(n)
