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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8004, base_ring=CyclotomicField(22))
M = H._module
chi = DirichletCharacter(H, M([0,11,19,0]))
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pari:[g,chi] = znchar(Mod(3365,8004))
\(\chi_{8004}(1625,\cdot)\)
\(\chi_{8004}(2321,\cdot)\)
\(\chi_{8004}(3365,\cdot)\)
\(\chi_{8004}(3713,\cdot)\)
\(\chi_{8004}(4757,\cdot)\)
\(\chi_{8004}(5801,\cdot)\)
\(\chi_{8004}(6497,\cdot)\)
\(\chi_{8004}(6845,\cdot)\)
\(\chi_{8004}(7193,\cdot)\)
\(\chi_{8004}(7541,\cdot)\)
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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 ]
\((4003,2669,3133,553)\) → \((1,-1,e\left(\frac{19}{22}\right),1)\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(25\) | \(31\) | \(35\) | \(37\) |
| \( \chi_{ 8004 }(3365, a) \) |
\(1\) | \(1\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{3}{22}\right)\) |
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sage:chi.jacobi_sum(n)
