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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(25410, base_ring=CyclotomicField(110))
M = H._module
chi = DirichletCharacter(H, M([0,0,55,53]))
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pari:[g,chi] = znchar(Mod(14671,25410))
\(\chi_{25410}(391,\cdot)\)
\(\chi_{25410}(601,\cdot)\)
\(\chi_{25410}(811,\cdot)\)
\(\chi_{25410}(1861,\cdot)\)
\(\chi_{25410}(2701,\cdot)\)
\(\chi_{25410}(2911,\cdot)\)
\(\chi_{25410}(3121,\cdot)\)
\(\chi_{25410}(4171,\cdot)\)
\(\chi_{25410}(5011,\cdot)\)
\(\chi_{25410}(5221,\cdot)\)
\(\chi_{25410}(5431,\cdot)\)
\(\chi_{25410}(6481,\cdot)\)
\(\chi_{25410}(7321,\cdot)\)
\(\chi_{25410}(7531,\cdot)\)
\(\chi_{25410}(8791,\cdot)\)
\(\chi_{25410}(9631,\cdot)\)
\(\chi_{25410}(10051,\cdot)\)
\(\chi_{25410}(11101,\cdot)\)
\(\chi_{25410}(11941,\cdot)\)
\(\chi_{25410}(12151,\cdot)\)
\(\chi_{25410}(12361,\cdot)\)
\(\chi_{25410}(13411,\cdot)\)
\(\chi_{25410}(14461,\cdot)\)
\(\chi_{25410}(14671,\cdot)\)
\(\chi_{25410}(16561,\cdot)\)
\(\chi_{25410}(16771,\cdot)\)
\(\chi_{25410}(16981,\cdot)\)
\(\chi_{25410}(18031,\cdot)\)
\(\chi_{25410}(18871,\cdot)\)
\(\chi_{25410}(19081,\cdot)\)
...
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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 ]
\((8471,15247,14521,7141)\) → \((1,1,-1,e\left(\frac{53}{110}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) | \(47\) |
| \( \chi_{ 25410 }(14671, a) \) |
\(1\) | \(1\) | \(e\left(\frac{9}{55}\right)\) | \(e\left(\frac{6}{55}\right)\) | \(e\left(\frac{27}{55}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{21}{110}\right)\) | \(e\left(\frac{103}{110}\right)\) | \(e\left(\frac{13}{55}\right)\) | \(e\left(\frac{32}{55}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{79}{110}\right)\) |
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sage:chi.jacobi_sum(n)
