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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5070, base_ring=CyclotomicField(78))
M = H._module
chi = DirichletCharacter(H, M([0,0,77]))
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pari:[g,chi] = znchar(Mod(1141,5070))
\(\chi_{5070}(121,\cdot)\)
\(\chi_{5070}(511,\cdot)\)
\(\chi_{5070}(751,\cdot)\)
\(\chi_{5070}(901,\cdot)\)
\(\chi_{5070}(1141,\cdot)\)
\(\chi_{5070}(1291,\cdot)\)
\(\chi_{5070}(1531,\cdot)\)
\(\chi_{5070}(1681,\cdot)\)
\(\chi_{5070}(1921,\cdot)\)
\(\chi_{5070}(2071,\cdot)\)
\(\chi_{5070}(2311,\cdot)\)
\(\chi_{5070}(2461,\cdot)\)
\(\chi_{5070}(2701,\cdot)\)
\(\chi_{5070}(3091,\cdot)\)
\(\chi_{5070}(3241,\cdot)\)
\(\chi_{5070}(3481,\cdot)\)
\(\chi_{5070}(3631,\cdot)\)
\(\chi_{5070}(3871,\cdot)\)
\(\chi_{5070}(4021,\cdot)\)
\(\chi_{5070}(4261,\cdot)\)
\(\chi_{5070}(4411,\cdot)\)
\(\chi_{5070}(4651,\cdot)\)
\(\chi_{5070}(4801,\cdot)\)
\(\chi_{5070}(5041,\cdot)\)
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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 ]
\((1691,4057,1861)\) → \((1,1,e\left(\frac{77}{78}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(7\) | \(11\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 5070 }(1141, a) \) |
\(1\) | \(1\) | \(e\left(\frac{49}{78}\right)\) | \(e\left(\frac{53}{78}\right)\) | \(e\left(\frac{5}{39}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{19}{39}\right)\) | \(e\left(\frac{19}{26}\right)\) | \(e\left(\frac{5}{78}\right)\) | \(e\left(\frac{71}{78}\right)\) | \(e\left(\frac{17}{39}\right)\) |
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sage:chi.jacobi_sum(n)
