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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9200, base_ring=CyclotomicField(110))
M = H._module
chi = DirichletCharacter(H, M([0,55,66,50]))
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pari:[g,chi] = znchar(Mod(3321,9200))
\(\chi_{9200}(41,\cdot)\)
\(\chi_{9200}(121,\cdot)\)
\(\chi_{9200}(361,\cdot)\)
\(\chi_{9200}(441,\cdot)\)
\(\chi_{9200}(761,\cdot)\)
\(\chi_{9200}(841,\cdot)\)
\(\chi_{9200}(1481,\cdot)\)
\(\chi_{9200}(1641,\cdot)\)
\(\chi_{9200}(1881,\cdot)\)
\(\chi_{9200}(1961,\cdot)\)
\(\chi_{9200}(2281,\cdot)\)
\(\chi_{9200}(2441,\cdot)\)
\(\chi_{9200}(2681,\cdot)\)
\(\chi_{9200}(2841,\cdot)\)
\(\chi_{9200}(3321,\cdot)\)
\(\chi_{9200}(3481,\cdot)\)
\(\chi_{9200}(3721,\cdot)\)
\(\chi_{9200}(4041,\cdot)\)
\(\chi_{9200}(4121,\cdot)\)
\(\chi_{9200}(4281,\cdot)\)
\(\chi_{9200}(4441,\cdot)\)
\(\chi_{9200}(4521,\cdot)\)
\(\chi_{9200}(4681,\cdot)\)
\(\chi_{9200}(5161,\cdot)\)
\(\chi_{9200}(5321,\cdot)\)
\(\chi_{9200}(5561,\cdot)\)
\(\chi_{9200}(5641,\cdot)\)
\(\chi_{9200}(5881,\cdot)\)
\(\chi_{9200}(5961,\cdot)\)
\(\chi_{9200}(6121,\cdot)\)
...
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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 ]
\((1151,6901,2577,1201)\) → \((1,-1,e\left(\frac{3}{5}\right),e\left(\frac{5}{11}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(27\) | \(29\) |
| \( \chi_{ 9200 }(3321, a) \) |
\(1\) | \(1\) | \(e\left(\frac{107}{110}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{52}{55}\right)\) | \(e\left(\frac{21}{110}\right)\) | \(e\left(\frac{29}{110}\right)\) | \(e\left(\frac{54}{55}\right)\) | \(e\left(\frac{13}{110}\right)\) | \(e\left(\frac{67}{110}\right)\) | \(e\left(\frac{101}{110}\right)\) | \(e\left(\frac{97}{110}\right)\) |
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sage:chi.jacobi_sum(n)
