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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([0,0,0,0,1]))
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pari:[g,chi] = znchar(Mod(12169,24648))
\(\chi_{24648}(937,\cdot)\)
\(\chi_{24648}(1561,\cdot)\)
\(\chi_{24648}(2497,\cdot)\)
\(\chi_{24648}(4057,\cdot)\)
\(\chi_{24648}(6553,\cdot)\)
\(\chi_{24648}(7489,\cdot)\)
\(\chi_{24648}(7801,\cdot)\)
\(\chi_{24648}(8737,\cdot)\)
\(\chi_{24648}(9049,\cdot)\)
\(\chi_{24648}(9361,\cdot)\)
\(\chi_{24648}(9673,\cdot)\)
\(\chi_{24648}(11857,\cdot)\)
\(\chi_{24648}(12169,\cdot)\)
\(\chi_{24648}(12793,\cdot)\)
\(\chi_{24648}(13105,\cdot)\)
\(\chi_{24648}(13417,\cdot)\)
\(\chi_{24648}(14353,\cdot)\)
\(\chi_{24648}(15913,\cdot)\)
\(\chi_{24648}(16225,\cdot)\)
\(\chi_{24648}(18097,\cdot)\)
\(\chi_{24648}(18721,\cdot)\)
\(\chi_{24648}(20593,\cdot)\)
\(\chi_{24648}(21841,\cdot)\)
\(\chi_{24648}(22465,\cdot)\)
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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 ]
\((18487,12325,16433,11377,12169)\) → \((1,1,1,1,e\left(\frac{1}{78}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 24648 }(12169, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{31}{39}\right)\) | \(e\left(\frac{53}{78}\right)\) | \(e\left(\frac{34}{39}\right)\) | \(e\left(\frac{7}{26}\right)\) | \(e\left(\frac{16}{39}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{23}{39}\right)\) | \(e\left(\frac{11}{78}\right)\) | \(e\left(\frac{28}{39}\right)\) | \(e\left(\frac{37}{78}\right)\) |
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sage:chi.jacobi_sum(n)
