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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8112, base_ring=CyclotomicField(156))
M = H._module
chi = DirichletCharacter(H, M([0,0,78,149]))
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pari:[g,chi] = znchar(Mod(305,8112))
\(\chi_{8112}(305,\cdot)\)
\(\chi_{8112}(353,\cdot)\)
\(\chi_{8112}(401,\cdot)\)
\(\chi_{8112}(449,\cdot)\)
\(\chi_{8112}(929,\cdot)\)
\(\chi_{8112}(977,\cdot)\)
\(\chi_{8112}(1025,\cdot)\)
\(\chi_{8112}(1073,\cdot)\)
\(\chi_{8112}(1553,\cdot)\)
\(\chi_{8112}(1649,\cdot)\)
\(\chi_{8112}(1697,\cdot)\)
\(\chi_{8112}(2177,\cdot)\)
\(\chi_{8112}(2225,\cdot)\)
\(\chi_{8112}(2273,\cdot)\)
\(\chi_{8112}(2321,\cdot)\)
\(\chi_{8112}(2801,\cdot)\)
\(\chi_{8112}(2849,\cdot)\)
\(\chi_{8112}(2897,\cdot)\)
\(\chi_{8112}(2945,\cdot)\)
\(\chi_{8112}(3425,\cdot)\)
\(\chi_{8112}(3473,\cdot)\)
\(\chi_{8112}(3521,\cdot)\)
\(\chi_{8112}(3569,\cdot)\)
\(\chi_{8112}(4049,\cdot)\)
\(\chi_{8112}(4097,\cdot)\)
\(\chi_{8112}(4193,\cdot)\)
\(\chi_{8112}(4673,\cdot)\)
\(\chi_{8112}(4721,\cdot)\)
\(\chi_{8112}(4769,\cdot)\)
\(\chi_{8112}(4817,\cdot)\)
...
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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 ]
\((5071,6085,2705,3889)\) → \((1,1,-1,e\left(\frac{149}{156}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 8112 }(305, a) \) |
\(1\) | \(1\) | \(e\left(\frac{5}{52}\right)\) | \(e\left(\frac{31}{156}\right)\) | \(e\left(\frac{137}{156}\right)\) | \(e\left(\frac{37}{39}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{5}{26}\right)\) | \(e\left(\frac{55}{78}\right)\) | \(e\left(\frac{3}{52}\right)\) | \(e\left(\frac{23}{78}\right)\) |
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sage:chi.jacobi_sum(n)
