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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(25410, base_ring=CyclotomicField(330))
M = H._module
chi = DirichletCharacter(H, M([165,0,110,243]))
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pari:[g,chi] = znchar(Mod(4391,25410))
\(\chi_{25410}(431,\cdot)\)
\(\chi_{25410}(611,\cdot)\)
\(\chi_{25410}(821,\cdot)\)
\(\chi_{25410}(1031,\cdot)\)
\(\chi_{25410}(1271,\cdot)\)
\(\chi_{25410}(1481,\cdot)\)
\(\chi_{25410}(2081,\cdot)\)
\(\chi_{25410}(2741,\cdot)\)
\(\chi_{25410}(2921,\cdot)\)
\(\chi_{25410}(3131,\cdot)\)
\(\chi_{25410}(3341,\cdot)\)
\(\chi_{25410}(3581,\cdot)\)
\(\chi_{25410}(4001,\cdot)\)
\(\chi_{25410}(4391,\cdot)\)
\(\chi_{25410}(5051,\cdot)\)
\(\chi_{25410}(5231,\cdot)\)
\(\chi_{25410}(5441,\cdot)\)
\(\chi_{25410}(5651,\cdot)\)
\(\chi_{25410}(5891,\cdot)\)
\(\chi_{25410}(6101,\cdot)\)
\(\chi_{25410}(6311,\cdot)\)
\(\chi_{25410}(6701,\cdot)\)
\(\chi_{25410}(7361,\cdot)\)
\(\chi_{25410}(7541,\cdot)\)
\(\chi_{25410}(7751,\cdot)\)
\(\chi_{25410}(7961,\cdot)\)
\(\chi_{25410}(8411,\cdot)\)
\(\chi_{25410}(8621,\cdot)\)
\(\chi_{25410}(9011,\cdot)\)
\(\chi_{25410}(9851,\cdot)\)
...
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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 ]
\((8471,15247,14521,7141)\) → \((-1,1,e\left(\frac{1}{3}\right),e\left(\frac{81}{110}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) | \(47\) |
| \( \chi_{ 25410 }(4391, a) \) |
\(1\) | \(1\) | \(e\left(\frac{41}{110}\right)\) | \(e\left(\frac{151}{165}\right)\) | \(e\left(\frac{259}{330}\right)\) | \(e\left(\frac{47}{66}\right)\) | \(e\left(\frac{1}{55}\right)\) | \(e\left(\frac{109}{165}\right)\) | \(e\left(\frac{98}{165}\right)\) | \(e\left(\frac{24}{55}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{109}{330}\right)\) |
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sage:chi.jacobi_sum(n)
