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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([98,0,37]))
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pari:[g,chi] = znchar(Mod(255,6304))
\(\chi_{6304}(31,\cdot)\)
\(\chi_{6304}(95,\cdot)\)
\(\chi_{6304}(159,\cdot)\)
\(\chi_{6304}(255,\cdot)\)
\(\chi_{6304}(319,\cdot)\)
\(\chi_{6304}(383,\cdot)\)
\(\chi_{6304}(415,\cdot)\)
\(\chi_{6304}(511,\cdot)\)
\(\chi_{6304}(543,\cdot)\)
\(\chi_{6304}(639,\cdot)\)
\(\chi_{6304}(671,\cdot)\)
\(\chi_{6304}(767,\cdot)\)
\(\chi_{6304}(799,\cdot)\)
\(\chi_{6304}(863,\cdot)\)
\(\chi_{6304}(927,\cdot)\)
\(\chi_{6304}(1023,\cdot)\)
\(\chi_{6304}(1087,\cdot)\)
\(\chi_{6304}(1151,\cdot)\)
\(\chi_{6304}(1503,\cdot)\)
\(\chi_{6304}(1695,\cdot)\)
\(\chi_{6304}(1727,\cdot)\)
\(\chi_{6304}(1791,\cdot)\)
\(\chi_{6304}(1823,\cdot)\)
\(\chi_{6304}(1855,\cdot)\)
\(\chi_{6304}(1983,\cdot)\)
\(\chi_{6304}(2015,\cdot)\)
\(\chi_{6304}(2111,\cdot)\)
\(\chi_{6304}(2175,\cdot)\)
\(\chi_{6304}(2239,\cdot)\)
\(\chi_{6304}(2367,\cdot)\)
...
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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 ]
\((1183,3941,3745)\) → \((-1,1,e\left(\frac{37}{196}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 6304 }(255, a) \) |
\(1\) | \(1\) | \(e\left(\frac{131}{196}\right)\) | \(e\left(\frac{157}{196}\right)\) | \(e\left(\frac{3}{49}\right)\) | \(e\left(\frac{33}{98}\right)\) | \(e\left(\frac{191}{196}\right)\) | \(e\left(\frac{141}{196}\right)\) | \(e\left(\frac{23}{49}\right)\) | \(e\left(\frac{3}{196}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{143}{196}\right)\) |
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sage:chi.jacobi_sum(n)
