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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(24648, base_ring=CyclotomicField(78))
M = H._module
chi = DirichletCharacter(H, M([39,39,0,13,49]))
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pari:[g,chi] = znchar(Mod(43,24648))
\(\chi_{24648}(43,\cdot)\)
\(\chi_{24648}(667,\cdot)\)
\(\chi_{24648}(3163,\cdot)\)
\(\chi_{24648}(3403,\cdot)\)
\(\chi_{24648}(4411,\cdot)\)
\(\chi_{24648}(5347,\cdot)\)
\(\chi_{24648}(5899,\cdot)\)
\(\chi_{24648}(6907,\cdot)\)
\(\chi_{24648}(7219,\cdot)\)
\(\chi_{24648}(7771,\cdot)\)
\(\chi_{24648}(9715,\cdot)\)
\(\chi_{24648}(10891,\cdot)\)
\(\chi_{24648}(11515,\cdot)\)
\(\chi_{24648}(12451,\cdot)\)
\(\chi_{24648}(12835,\cdot)\)
\(\chi_{24648}(16507,\cdot)\)
\(\chi_{24648}(18691,\cdot)\)
\(\chi_{24648}(19315,\cdot)\)
\(\chi_{24648}(19699,\cdot)\)
\(\chi_{24648}(21811,\cdot)\)
\(\chi_{24648}(22747,\cdot)\)
\(\chi_{24648}(23059,\cdot)\)
\(\chi_{24648}(23131,\cdot)\)
\(\chi_{24648}(23443,\cdot)\)
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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 ]
\((18487,12325,16433,11377,12169)\) → \((-1,-1,1,e\left(\frac{1}{6}\right),e\left(\frac{49}{78}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 24648 }(43, a) \) |
\(1\) | \(1\) | \(e\left(\frac{37}{39}\right)\) | \(e\left(\frac{49}{78}\right)\) | \(e\left(\frac{23}{26}\right)\) | \(e\left(\frac{41}{78}\right)\) | \(e\left(\frac{73}{78}\right)\) | \(-1\) | \(e\left(\frac{35}{39}\right)\) | \(e\left(\frac{1}{13}\right)\) | \(e\left(\frac{7}{39}\right)\) | \(e\left(\frac{15}{26}\right)\) |
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sage:chi.jacobi_sum(n)
