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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1900, base_ring=CyclotomicField(90))
M = H._module
chi = DirichletCharacter(H, M([0,54,50]))
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pari:[g,chi] = znchar(Mod(321,1900))
\(\chi_{1900}(61,\cdot)\)
\(\chi_{1900}(81,\cdot)\)
\(\chi_{1900}(161,\cdot)\)
\(\chi_{1900}(321,\cdot)\)
\(\chi_{1900}(441,\cdot)\)
\(\chi_{1900}(461,\cdot)\)
\(\chi_{1900}(481,\cdot)\)
\(\chi_{1900}(541,\cdot)\)
\(\chi_{1900}(681,\cdot)\)
\(\chi_{1900}(821,\cdot)\)
\(\chi_{1900}(841,\cdot)\)
\(\chi_{1900}(861,\cdot)\)
\(\chi_{1900}(921,\cdot)\)
\(\chi_{1900}(1061,\cdot)\)
\(\chi_{1900}(1081,\cdot)\)
\(\chi_{1900}(1221,\cdot)\)
\(\chi_{1900}(1241,\cdot)\)
\(\chi_{1900}(1441,\cdot)\)
\(\chi_{1900}(1461,\cdot)\)
\(\chi_{1900}(1581,\cdot)\)
\(\chi_{1900}(1621,\cdot)\)
\(\chi_{1900}(1681,\cdot)\)
\(\chi_{1900}(1821,\cdot)\)
\(\chi_{1900}(1841,\cdot)\)
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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 ]
\((951,77,401)\) → \((1,e\left(\frac{3}{5}\right),e\left(\frac{5}{9}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 1900 }(321, a) \) |
\(1\) | \(1\) | \(e\left(\frac{19}{45}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{38}{45}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{8}{45}\right)\) | \(e\left(\frac{16}{45}\right)\) | \(e\left(\frac{34}{45}\right)\) | \(e\left(\frac{32}{45}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{29}{45}\right)\) |
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sage:chi.jacobi_sum(n)
