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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3800, base_ring=CyclotomicField(180))
M = H._module
chi = DirichletCharacter(H, M([90,0,99,20]))
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pari:[g,chi] = znchar(Mod(23,3800))
\(\chi_{3800}(23,\cdot)\)
\(\chi_{3800}(47,\cdot)\)
\(\chi_{3800}(63,\cdot)\)
\(\chi_{3800}(263,\cdot)\)
\(\chi_{3800}(327,\cdot)\)
\(\chi_{3800}(367,\cdot)\)
\(\chi_{3800}(423,\cdot)\)
\(\chi_{3800}(503,\cdot)\)
\(\chi_{3800}(567,\cdot)\)
\(\chi_{3800}(663,\cdot)\)
\(\chi_{3800}(727,\cdot)\)
\(\chi_{3800}(783,\cdot)\)
\(\chi_{3800}(823,\cdot)\)
\(\chi_{3800}(967,\cdot)\)
\(\chi_{3800}(1023,\cdot)\)
\(\chi_{3800}(1087,\cdot)\)
\(\chi_{3800}(1127,\cdot)\)
\(\chi_{3800}(1183,\cdot)\)
\(\chi_{3800}(1263,\cdot)\)
\(\chi_{3800}(1327,\cdot)\)
\(\chi_{3800}(1423,\cdot)\)
\(\chi_{3800}(1487,\cdot)\)
\(\chi_{3800}(1567,\cdot)\)
\(\chi_{3800}(1583,\cdot)\)
\(\chi_{3800}(1727,\cdot)\)
\(\chi_{3800}(1783,\cdot)\)
\(\chi_{3800}(1847,\cdot)\)
\(\chi_{3800}(1887,\cdot)\)
\(\chi_{3800}(2023,\cdot)\)
\(\chi_{3800}(2087,\cdot)\)
...
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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 ]
\((951,1901,1977,401)\) → \((-1,1,e\left(\frac{11}{20}\right),e\left(\frac{1}{9}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 3800 }(23, a) \) |
\(1\) | \(1\) | \(e\left(\frac{143}{180}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{53}{90}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{1}{180}\right)\) | \(e\left(\frac{47}{180}\right)\) | \(e\left(\frac{32}{45}\right)\) | \(e\left(\frac{139}{180}\right)\) | \(e\left(\frac{23}{60}\right)\) | \(e\left(\frac{89}{90}\right)\) |
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sage:chi.jacobi_sum(n)
