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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([0,70,133,100]))
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pari:[g,chi] = znchar(Mod(113,14700))
\(\chi_{14700}(113,\cdot)\)
\(\chi_{14700}(533,\cdot)\)
\(\chi_{14700}(617,\cdot)\)
\(\chi_{14700}(953,\cdot)\)
\(\chi_{14700}(1037,\cdot)\)
\(\chi_{14700}(1877,\cdot)\)
\(\chi_{14700}(2213,\cdot)\)
\(\chi_{14700}(2297,\cdot)\)
\(\chi_{14700}(2633,\cdot)\)
\(\chi_{14700}(2717,\cdot)\)
\(\chi_{14700}(3053,\cdot)\)
\(\chi_{14700}(3473,\cdot)\)
\(\chi_{14700}(3977,\cdot)\)
\(\chi_{14700}(4397,\cdot)\)
\(\chi_{14700}(4733,\cdot)\)
\(\chi_{14700}(4817,\cdot)\)
\(\chi_{14700}(5153,\cdot)\)
\(\chi_{14700}(5237,\cdot)\)
\(\chi_{14700}(5573,\cdot)\)
\(\chi_{14700}(6413,\cdot)\)
\(\chi_{14700}(6497,\cdot)\)
\(\chi_{14700}(6833,\cdot)\)
\(\chi_{14700}(6917,\cdot)\)
\(\chi_{14700}(7337,\cdot)\)
\(\chi_{14700}(7673,\cdot)\)
\(\chi_{14700}(8177,\cdot)\)
\(\chi_{14700}(8513,\cdot)\)
\(\chi_{14700}(8597,\cdot)\)
\(\chi_{14700}(8933,\cdot)\)
\(\chi_{14700}(9353,\cdot)\)
...
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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 ]
\((7351,4901,1177,9901)\) → \((1,-1,e\left(\frac{19}{20}\right),e\left(\frac{5}{7}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 14700 }(113, a) \) |
\(1\) | \(1\) | \(e\left(\frac{19}{70}\right)\) | \(e\left(\frac{87}{140}\right)\) | \(e\left(\frac{99}{140}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{13}{140}\right)\) | \(e\left(\frac{9}{35}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{57}{140}\right)\) | \(e\left(\frac{1}{70}\right)\) | \(e\left(\frac{15}{28}\right)\) |
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sage:chi.jacobi_sum(n)
