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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1280, base_ring=CyclotomicField(32))
M = H._module
chi = DirichletCharacter(H, M([0,9,16]))
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pari:[g,chi] = znchar(Mod(729,1280))
\(\chi_{1280}(9,\cdot)\)
\(\chi_{1280}(89,\cdot)\)
\(\chi_{1280}(169,\cdot)\)
\(\chi_{1280}(249,\cdot)\)
\(\chi_{1280}(329,\cdot)\)
\(\chi_{1280}(409,\cdot)\)
\(\chi_{1280}(489,\cdot)\)
\(\chi_{1280}(569,\cdot)\)
\(\chi_{1280}(649,\cdot)\)
\(\chi_{1280}(729,\cdot)\)
\(\chi_{1280}(809,\cdot)\)
\(\chi_{1280}(889,\cdot)\)
\(\chi_{1280}(969,\cdot)\)
\(\chi_{1280}(1049,\cdot)\)
\(\chi_{1280}(1129,\cdot)\)
\(\chi_{1280}(1209,\cdot)\)
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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 ]
\((511,261,257)\) → \((1,e\left(\frac{9}{32}\right),-1)\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
| \( \chi_{ 1280 }(729, a) \) |
\(1\) | \(1\) | \(e\left(\frac{11}{32}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{29}{32}\right)\) | \(e\left(\frac{23}{32}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{15}{32}\right)\) | \(e\left(\frac{21}{32}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{1}{32}\right)\) |
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sage:chi.jacobi_sum(n)
