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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8820, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([0,14,0,18]))
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pari:[g,chi] = znchar(Mod(3361,8820))
\(\chi_{8820}(421,\cdot)\)
\(\chi_{8820}(841,\cdot)\)
\(\chi_{8820}(1681,\cdot)\)
\(\chi_{8820}(2101,\cdot)\)
\(\chi_{8820}(3361,\cdot)\)
\(\chi_{8820}(4201,\cdot)\)
\(\chi_{8820}(4621,\cdot)\)
\(\chi_{8820}(5461,\cdot)\)
\(\chi_{8820}(6721,\cdot)\)
\(\chi_{8820}(7141,\cdot)\)
\(\chi_{8820}(7981,\cdot)\)
\(\chi_{8820}(8401,\cdot)\)
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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 ]
\((4411,7841,7057,1081)\) → \((1,e\left(\frac{1}{3}\right),1,e\left(\frac{3}{7}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 8820 }(3361, a) \) |
\(1\) | \(1\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(1\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{19}{21}\right)\) |
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sage:chi.jacobi_sum(n)
