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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5408, base_ring=CyclotomicField(52))
M = H._module
chi = DirichletCharacter(H, M([0,39,36]))
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pari:[g,chi] = znchar(Mod(521,5408))
\(\chi_{5408}(105,\cdot)\)
\(\chi_{5408}(313,\cdot)\)
\(\chi_{5408}(521,\cdot)\)
\(\chi_{5408}(729,\cdot)\)
\(\chi_{5408}(937,\cdot)\)
\(\chi_{5408}(1145,\cdot)\)
\(\chi_{5408}(1561,\cdot)\)
\(\chi_{5408}(1769,\cdot)\)
\(\chi_{5408}(1977,\cdot)\)
\(\chi_{5408}(2185,\cdot)\)
\(\chi_{5408}(2393,\cdot)\)
\(\chi_{5408}(2601,\cdot)\)
\(\chi_{5408}(2809,\cdot)\)
\(\chi_{5408}(3017,\cdot)\)
\(\chi_{5408}(3225,\cdot)\)
\(\chi_{5408}(3433,\cdot)\)
\(\chi_{5408}(3641,\cdot)\)
\(\chi_{5408}(3849,\cdot)\)
\(\chi_{5408}(4265,\cdot)\)
\(\chi_{5408}(4473,\cdot)\)
\(\chi_{5408}(4681,\cdot)\)
\(\chi_{5408}(4889,\cdot)\)
\(\chi_{5408}(5097,\cdot)\)
\(\chi_{5408}(5305,\cdot)\)
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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 ]
\((2367,677,1185)\) → \((1,-i,e\left(\frac{9}{13}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(15\) | \(17\) | \(19\) | \(21\) | \(23\) |
| \( \chi_{ 5408 }(521, a) \) |
\(1\) | \(1\) | \(e\left(\frac{5}{52}\right)\) | \(e\left(\frac{51}{52}\right)\) | \(e\left(\frac{15}{26}\right)\) | \(e\left(\frac{5}{26}\right)\) | \(e\left(\frac{3}{52}\right)\) | \(e\left(\frac{1}{13}\right)\) | \(e\left(\frac{1}{13}\right)\) | \(i\) | \(e\left(\frac{35}{52}\right)\) | \(-1\) |
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sage:chi.jacobi_sum(n)
