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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(14700, base_ring=CyclotomicField(210))
M = H._module
chi = DirichletCharacter(H, M([0,105,63,115]))
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pari:[g,chi] = znchar(Mod(89,14700))
\(\chi_{14700}(89,\cdot)\)
\(\chi_{14700}(269,\cdot)\)
\(\chi_{14700}(689,\cdot)\)
\(\chi_{14700}(929,\cdot)\)
\(\chi_{14700}(1529,\cdot)\)
\(\chi_{14700}(1769,\cdot)\)
\(\chi_{14700}(2189,\cdot)\)
\(\chi_{14700}(2369,\cdot)\)
\(\chi_{14700}(2609,\cdot)\)
\(\chi_{14700}(2789,\cdot)\)
\(\chi_{14700}(3029,\cdot)\)
\(\chi_{14700}(3209,\cdot)\)
\(\chi_{14700}(3629,\cdot)\)
\(\chi_{14700}(3869,\cdot)\)
\(\chi_{14700}(4289,\cdot)\)
\(\chi_{14700}(4469,\cdot)\)
\(\chi_{14700}(4709,\cdot)\)
\(\chi_{14700}(4889,\cdot)\)
\(\chi_{14700}(5129,\cdot)\)
\(\chi_{14700}(5309,\cdot)\)
\(\chi_{14700}(5729,\cdot)\)
\(\chi_{14700}(5969,\cdot)\)
\(\chi_{14700}(6569,\cdot)\)
\(\chi_{14700}(6809,\cdot)\)
\(\chi_{14700}(7229,\cdot)\)
\(\chi_{14700}(7409,\cdot)\)
\(\chi_{14700}(7829,\cdot)\)
\(\chi_{14700}(8069,\cdot)\)
\(\chi_{14700}(8489,\cdot)\)
\(\chi_{14700}(8669,\cdot)\)
...
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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 ]
\((7351,4901,1177,9901)\) → \((1,-1,e\left(\frac{3}{10}\right),e\left(\frac{23}{42}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 14700 }(89, a) \) |
\(1\) | \(1\) | \(e\left(\frac{43}{210}\right)\) | \(e\left(\frac{27}{35}\right)\) | \(e\left(\frac{19}{210}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{64}{105}\right)\) | \(e\left(\frac{67}{70}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{47}{210}\right)\) | \(e\left(\frac{32}{35}\right)\) | \(e\left(\frac{11}{14}\right)\) |
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sage:chi.jacobi_sum(n)
