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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(14700, base_ring=CyclotomicField(70))
M = H._module
chi = DirichletCharacter(H, M([35,0,21,25]))
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pari:[g,chi] = znchar(Mod(139,14700))
\(\chi_{14700}(139,\cdot)\)
\(\chi_{14700}(559,\cdot)\)
\(\chi_{14700}(1819,\cdot)\)
\(\chi_{14700}(2239,\cdot)\)
\(\chi_{14700}(2659,\cdot)\)
\(\chi_{14700}(3079,\cdot)\)
\(\chi_{14700}(4339,\cdot)\)
\(\chi_{14700}(4759,\cdot)\)
\(\chi_{14700}(5179,\cdot)\)
\(\chi_{14700}(6019,\cdot)\)
\(\chi_{14700}(6439,\cdot)\)
\(\chi_{14700}(7279,\cdot)\)
\(\chi_{14700}(8119,\cdot)\)
\(\chi_{14700}(8539,\cdot)\)
\(\chi_{14700}(8959,\cdot)\)
\(\chi_{14700}(9379,\cdot)\)
\(\chi_{14700}(10219,\cdot)\)
\(\chi_{14700}(10639,\cdot)\)
\(\chi_{14700}(11059,\cdot)\)
\(\chi_{14700}(11479,\cdot)\)
\(\chi_{14700}(12319,\cdot)\)
\(\chi_{14700}(13159,\cdot)\)
\(\chi_{14700}(13579,\cdot)\)
\(\chi_{14700}(14419,\cdot)\)
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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 ]
\((7351,4901,1177,9901)\) → \((-1,1,e\left(\frac{3}{10}\right),e\left(\frac{5}{14}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 14700 }(139, a) \) |
\(1\) | \(1\) | \(e\left(\frac{41}{70}\right)\) | \(e\left(\frac{17}{35}\right)\) | \(e\left(\frac{29}{35}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{13}{35}\right)\) | \(e\left(\frac{1}{35}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{9}{70}\right)\) | \(e\left(\frac{39}{70}\right)\) | \(e\left(\frac{1}{7}\right)\) |
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sage:chi.jacobi_sum(n)
