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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(24648, base_ring=CyclotomicField(156))
M = H._module
chi = DirichletCharacter(H, M([78,78,0,91,116]))
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pari:[g,chi] = znchar(Mod(11971,24648))
\(\chi_{24648}(19,\cdot)\)
\(\chi_{24648}(115,\cdot)\)
\(\chi_{24648}(163,\cdot)\)
\(\chi_{24648}(643,\cdot)\)
\(\chi_{24648}(1099,\cdot)\)
\(\chi_{24648}(1315,\cdot)\)
\(\chi_{24648}(1363,\cdot)\)
\(\chi_{24648}(1675,\cdot)\)
\(\chi_{24648}(3595,\cdot)\)
\(\chi_{24648}(3763,\cdot)\)
\(\chi_{24648}(4219,\cdot)\)
\(\chi_{24648}(5107,\cdot)\)
\(\chi_{24648}(5635,\cdot)\)
\(\chi_{24648}(6403,\cdot)\)
\(\chi_{24648}(8131,\cdot)\)
\(\chi_{24648}(8179,\cdot)\)
\(\chi_{24648}(10459,\cdot)\)
\(\chi_{24648}(11299,\cdot)\)
\(\chi_{24648}(11395,\cdot)\)
\(\chi_{24648}(11971,\cdot)\)
\(\chi_{24648}(12019,\cdot)\)
\(\chi_{24648}(13123,\cdot)\)
\(\chi_{24648}(13435,\cdot)\)
\(\chi_{24648}(13795,\cdot)\)
\(\chi_{24648}(14107,\cdot)\)
\(\chi_{24648}(14371,\cdot)\)
\(\chi_{24648}(15091,\cdot)\)
\(\chi_{24648}(15139,\cdot)\)
\(\chi_{24648}(15667,\cdot)\)
\(\chi_{24648}(16603,\cdot)\)
...
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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 ]
\((18487,12325,16433,11377,12169)\) → \((-1,-1,1,e\left(\frac{7}{12}\right),e\left(\frac{29}{39}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 24648 }(11971, a) \) |
\(1\) | \(1\) | \(e\left(\frac{133}{156}\right)\) | \(e\left(\frac{17}{52}\right)\) | \(e\left(\frac{101}{156}\right)\) | \(e\left(\frac{61}{78}\right)\) | \(e\left(\frac{37}{52}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{55}{78}\right)\) | \(e\left(\frac{1}{78}\right)\) | \(e\left(\frac{61}{156}\right)\) | \(e\left(\frac{7}{39}\right)\) |
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sage:chi.jacobi_sum(n)
