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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([78,0,0,125]))
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pari:[g,chi] = znchar(Mod(175,8112))
\(\chi_{8112}(175,\cdot)\)
\(\chi_{8112}(223,\cdot)\)
\(\chi_{8112}(271,\cdot)\)
\(\chi_{8112}(799,\cdot)\)
\(\chi_{8112}(847,\cdot)\)
\(\chi_{8112}(895,\cdot)\)
\(\chi_{8112}(943,\cdot)\)
\(\chi_{8112}(1423,\cdot)\)
\(\chi_{8112}(1471,\cdot)\)
\(\chi_{8112}(1519,\cdot)\)
\(\chi_{8112}(1567,\cdot)\)
\(\chi_{8112}(2095,\cdot)\)
\(\chi_{8112}(2143,\cdot)\)
\(\chi_{8112}(2191,\cdot)\)
\(\chi_{8112}(2671,\cdot)\)
\(\chi_{8112}(2719,\cdot)\)
\(\chi_{8112}(2767,\cdot)\)
\(\chi_{8112}(2815,\cdot)\)
\(\chi_{8112}(3295,\cdot)\)
\(\chi_{8112}(3343,\cdot)\)
\(\chi_{8112}(3391,\cdot)\)
\(\chi_{8112}(3439,\cdot)\)
\(\chi_{8112}(3919,\cdot)\)
\(\chi_{8112}(4015,\cdot)\)
\(\chi_{8112}(4063,\cdot)\)
\(\chi_{8112}(4543,\cdot)\)
\(\chi_{8112}(4591,\cdot)\)
\(\chi_{8112}(4639,\cdot)\)
\(\chi_{8112}(4687,\cdot)\)
\(\chi_{8112}(5167,\cdot)\)
...
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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 ]
\((5071,6085,2705,3889)\) → \((-1,1,1,e\left(\frac{125}{156}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 8112 }(175, a) \) |
\(1\) | \(1\) | \(e\left(\frac{11}{52}\right)\) | \(e\left(\frac{37}{156}\right)\) | \(e\left(\frac{5}{156}\right)\) | \(e\left(\frac{77}{78}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{11}{26}\right)\) | \(e\left(\frac{2}{39}\right)\) | \(e\left(\frac{17}{52}\right)\) | \(e\left(\frac{35}{78}\right)\) |
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sage:chi.jacobi_sum(n)
