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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([0,98,137]))
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pari:[g,chi] = znchar(Mod(753,6304))
\(\chi_{6304}(17,\cdot)\)
\(\chi_{6304}(145,\cdot)\)
\(\chi_{6304}(209,\cdot)\)
\(\chi_{6304}(241,\cdot)\)
\(\chi_{6304}(305,\cdot)\)
\(\chi_{6304}(337,\cdot)\)
\(\chi_{6304}(465,\cdot)\)
\(\chi_{6304}(497,\cdot)\)
\(\chi_{6304}(561,\cdot)\)
\(\chi_{6304}(593,\cdot)\)
\(\chi_{6304}(657,\cdot)\)
\(\chi_{6304}(689,\cdot)\)
\(\chi_{6304}(721,\cdot)\)
\(\chi_{6304}(753,\cdot)\)
\(\chi_{6304}(785,\cdot)\)
\(\chi_{6304}(913,\cdot)\)
\(\chi_{6304}(977,\cdot)\)
\(\chi_{6304}(1041,\cdot)\)
\(\chi_{6304}(1137,\cdot)\)
\(\chi_{6304}(1169,\cdot)\)
\(\chi_{6304}(1297,\cdot)\)
\(\chi_{6304}(1329,\cdot)\)
\(\chi_{6304}(1361,\cdot)\)
\(\chi_{6304}(1425,\cdot)\)
\(\chi_{6304}(1457,\cdot)\)
\(\chi_{6304}(1649,\cdot)\)
\(\chi_{6304}(2001,\cdot)\)
\(\chi_{6304}(2065,\cdot)\)
\(\chi_{6304}(2129,\cdot)\)
\(\chi_{6304}(2225,\cdot)\)
...
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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 ]
\((1183,3941,3745)\) → \((1,-1,e\left(\frac{137}{196}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 6304 }(753, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{3}{196}\right)\) | \(e\left(\frac{139}{196}\right)\) | \(e\left(\frac{5}{98}\right)\) | \(e\left(\frac{3}{98}\right)\) | \(e\left(\frac{151}{196}\right)\) | \(e\left(\frac{191}{196}\right)\) | \(e\left(\frac{71}{98}\right)\) | \(e\left(\frac{27}{196}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{13}{196}\right)\) |
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sage:chi.jacobi_sum(n)
