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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3344, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([0,27,0,4]))
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pari:[g,chi] = znchar(Mod(1277,3344))
\(\chi_{3344}(309,\cdot)\)
\(\chi_{3344}(397,\cdot)\)
\(\chi_{3344}(1013,\cdot)\)
\(\chi_{3344}(1277,\cdot)\)
\(\chi_{3344}(1365,\cdot)\)
\(\chi_{3344}(1453,\cdot)\)
\(\chi_{3344}(1981,\cdot)\)
\(\chi_{3344}(2069,\cdot)\)
\(\chi_{3344}(2685,\cdot)\)
\(\chi_{3344}(2949,\cdot)\)
\(\chi_{3344}(3037,\cdot)\)
\(\chi_{3344}(3125,\cdot)\)
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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 ]
\((2927,837,2433,705)\) → \((1,-i,1,e\left(\frac{1}{9}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(13\) | \(15\) | \(17\) | \(21\) | \(23\) | \(25\) |
| \( \chi_{ 3344 }(1277, a) \) |
\(1\) | \(1\) | \(e\left(\frac{25}{36}\right)\) | \(e\left(\frac{19}{36}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{29}{36}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{1}{18}\right)\) |
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sage:chi.jacobi_sum(n)
