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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6304, base_ring=CyclotomicField(196))
M = H._module
chi = DirichletCharacter(H, M([0,49,74]))
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pari:[g,chi] = znchar(Mod(409,6304))
\(\chi_{6304}(9,\cdot)\)
\(\chi_{6304}(25,\cdot)\)
\(\chi_{6304}(41,\cdot)\)
\(\chi_{6304}(121,\cdot)\)
\(\chi_{6304}(137,\cdot)\)
\(\chi_{6304}(169,\cdot)\)
\(\chi_{6304}(201,\cdot)\)
\(\chi_{6304}(313,\cdot)\)
\(\chi_{6304}(345,\cdot)\)
\(\chi_{6304}(409,\cdot)\)
\(\chi_{6304}(441,\cdot)\)
\(\chi_{6304}(521,\cdot)\)
\(\chi_{6304}(537,\cdot)\)
\(\chi_{6304}(601,\cdot)\)
\(\chi_{6304}(617,\cdot)\)
\(\chi_{6304}(729,\cdot)\)
\(\chi_{6304}(969,\cdot)\)
\(\chi_{6304}(1049,\cdot)\)
\(\chi_{6304}(1081,\cdot)\)
\(\chi_{6304}(1097,\cdot)\)
\(\chi_{6304}(1129,\cdot)\)
\(\chi_{6304}(1145,\cdot)\)
\(\chi_{6304}(1225,\cdot)\)
\(\chi_{6304}(1289,\cdot)\)
\(\chi_{6304}(1337,\cdot)\)
\(\chi_{6304}(1401,\cdot)\)
\(\chi_{6304}(1513,\cdot)\)
\(\chi_{6304}(1641,\cdot)\)
\(\chi_{6304}(1673,\cdot)\)
\(\chi_{6304}(1865,\cdot)\)
...
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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 ]
\((1183,3941,3745)\) → \((1,i,e\left(\frac{37}{98}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 6304 }(409, a) \) |
\(1\) | \(1\) | \(e\left(\frac{17}{196}\right)\) | \(e\left(\frac{167}{196}\right)\) | \(e\left(\frac{61}{98}\right)\) | \(e\left(\frac{17}{98}\right)\) | \(e\left(\frac{39}{196}\right)\) | \(e\left(\frac{37}{196}\right)\) | \(e\left(\frac{46}{49}\right)\) | \(e\left(\frac{3}{98}\right)\) | \(e\left(\frac{25}{28}\right)\) | \(e\left(\frac{139}{196}\right)\) |
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sage:chi.jacobi_sum(n)
