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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6336, base_ring=CyclotomicField(80))
M = H._module
chi = DirichletCharacter(H, M([40,5,40,48]))
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pari:[g,chi] = znchar(Mod(251,6336))
\(\chi_{6336}(179,\cdot)\)
\(\chi_{6336}(251,\cdot)\)
\(\chi_{6336}(323,\cdot)\)
\(\chi_{6336}(467,\cdot)\)
\(\chi_{6336}(971,\cdot)\)
\(\chi_{6336}(1043,\cdot)\)
\(\chi_{6336}(1115,\cdot)\)
\(\chi_{6336}(1259,\cdot)\)
\(\chi_{6336}(1763,\cdot)\)
\(\chi_{6336}(1835,\cdot)\)
\(\chi_{6336}(1907,\cdot)\)
\(\chi_{6336}(2051,\cdot)\)
\(\chi_{6336}(2555,\cdot)\)
\(\chi_{6336}(2627,\cdot)\)
\(\chi_{6336}(2699,\cdot)\)
\(\chi_{6336}(2843,\cdot)\)
\(\chi_{6336}(3347,\cdot)\)
\(\chi_{6336}(3419,\cdot)\)
\(\chi_{6336}(3491,\cdot)\)
\(\chi_{6336}(3635,\cdot)\)
\(\chi_{6336}(4139,\cdot)\)
\(\chi_{6336}(4211,\cdot)\)
\(\chi_{6336}(4283,\cdot)\)
\(\chi_{6336}(4427,\cdot)\)
\(\chi_{6336}(4931,\cdot)\)
\(\chi_{6336}(5003,\cdot)\)
\(\chi_{6336}(5075,\cdot)\)
\(\chi_{6336}(5219,\cdot)\)
\(\chi_{6336}(5723,\cdot)\)
\(\chi_{6336}(5795,\cdot)\)
...
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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 ]
\((4159,4357,3521,1729)\) → \((-1,e\left(\frac{1}{16}\right),-1,e\left(\frac{3}{5}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 6336 }(251, a) \) |
\(1\) | \(1\) | \(e\left(\frac{77}{80}\right)\) | \(e\left(\frac{13}{40}\right)\) | \(e\left(\frac{43}{80}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{59}{80}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{31}{80}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{23}{80}\right)\) |
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sage:chi.jacobi_sum(n)
