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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(309680, base_ring=CyclotomicField(1092))
M = H._module
chi = DirichletCharacter(H, M([546,819,0,832,392]))
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pari:[g,chi] = znchar(Mod(6211,309680))
\(\chi_{309680}(11,\cdot)\)
\(\chi_{309680}(51,\cdot)\)
\(\chi_{309680}(1691,\cdot)\)
\(\chi_{309680}(2011,\cdot)\)
\(\chi_{309680}(5651,\cdot)\)
\(\chi_{309680}(6171,\cdot)\)
\(\chi_{309680}(6211,\cdot)\)
\(\chi_{309680}(6491,\cdot)\)
\(\chi_{309680}(7051,\cdot)\)
\(\chi_{309680}(8131,\cdot)\)
\(\chi_{309680}(11211,\cdot)\)
\(\chi_{309680}(11531,\cdot)\)
\(\chi_{309680}(13771,\cdot)\)
\(\chi_{309680}(18251,\cdot)\)
\(\chi_{309680}(18531,\cdot)\)
\(\chi_{309680}(19371,\cdot)\)
\(\chi_{309680}(19611,\cdot)\)
\(\chi_{309680}(20171,\cdot)\)
\(\chi_{309680}(21291,\cdot)\)
\(\chi_{309680}(21571,\cdot)\)
\(\chi_{309680}(21611,\cdot)\)
\(\chi_{309680}(22131,\cdot)\)
\(\chi_{309680}(22171,\cdot)\)
\(\chi_{309680}(23811,\cdot)\)
\(\chi_{309680}(24131,\cdot)\)
\(\chi_{309680}(26891,\cdot)\)
\(\chi_{309680}(27771,\cdot)\)
\(\chi_{309680}(28331,\cdot)\)
\(\chi_{309680}(28611,\cdot)\)
\(\chi_{309680}(29171,\cdot)\)
...
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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 ]
\((193551,232261,61937,297041,82321)\) → \((-1,-i,1,e\left(\frac{16}{21}\right),e\left(\frac{14}{39}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(27\) | \(29\) | \(31\) |
| \( \chi_{ 309680 }(6211, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{317}{364}\right)\) | \(e\left(\frac{135}{182}\right)\) | \(e\left(\frac{149}{1092}\right)\) | \(e\left(\frac{653}{1092}\right)\) | \(e\left(\frac{160}{273}\right)\) | \(e\left(\frac{47}{52}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{223}{364}\right)\) | \(e\left(\frac{997}{1092}\right)\) | \(e\left(\frac{73}{78}\right)\) |
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sage:chi.jacobi_sum(n)
