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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([0,39,0,26,15]))
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pari:[g,chi] = znchar(Mod(24469,24648))
\(\chi_{24648}(61,\cdot)\)
\(\chi_{24648}(373,\cdot)\)
\(\chi_{24648}(1621,\cdot)\)
\(\chi_{24648}(2245,\cdot)\)
\(\chi_{24648}(3253,\cdot)\)
\(\chi_{24648}(3493,\cdot)\)
\(\chi_{24648}(5125,\cdot)\)
\(\chi_{24648}(5749,\cdot)\)
\(\chi_{24648}(6061,\cdot)\)
\(\chi_{24648}(7309,\cdot)\)
\(\chi_{24648}(7933,\cdot)\)
\(\chi_{24648}(9181,\cdot)\)
\(\chi_{24648}(10045,\cdot)\)
\(\chi_{24648}(11605,\cdot)\)
\(\chi_{24648}(15037,\cdot)\)
\(\chi_{24648}(15733,\cdot)\)
\(\chi_{24648}(15973,\cdot)\)
\(\chi_{24648}(17293,\cdot)\)
\(\chi_{24648}(18781,\cdot)\)
\(\chi_{24648}(20725,\cdot)\)
\(\chi_{24648}(21661,\cdot)\)
\(\chi_{24648}(22213,\cdot)\)
\(\chi_{24648}(24085,\cdot)\)
\(\chi_{24648}(24469,\cdot)\)
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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 ]
\((18487,12325,16433,11377,12169)\) → \((1,-1,1,e\left(\frac{1}{3}\right),e\left(\frac{5}{26}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 24648 }(24469, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{11}{26}\right)\) | \(e\left(\frac{67}{78}\right)\) | \(e\left(\frac{71}{78}\right)\) | \(e\left(\frac{55}{78}\right)\) | \(e\left(\frac{25}{78}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{37}{39}\right)\) | \(e\left(\frac{10}{13}\right)\) | \(e\left(\frac{11}{39}\right)\) |
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sage:chi.jacobi_sum(n)
