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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3400, base_ring=CyclotomicField(40))
M = H._module
chi = DirichletCharacter(H, M([0,0,28,5]))
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pari:[g,chi] = znchar(Mod(9,3400))
\(\chi_{3400}(9,\cdot)\)
\(\chi_{3400}(529,\cdot)\)
\(\chi_{3400}(569,\cdot)\)
\(\chi_{3400}(689,\cdot)\)
\(\chi_{3400}(729,\cdot)\)
\(\chi_{3400}(1209,\cdot)\)
\(\chi_{3400}(1369,\cdot)\)
\(\chi_{3400}(1409,\cdot)\)
\(\chi_{3400}(1889,\cdot)\)
\(\chi_{3400}(1929,\cdot)\)
\(\chi_{3400}(2089,\cdot)\)
\(\chi_{3400}(2569,\cdot)\)
\(\chi_{3400}(2609,\cdot)\)
\(\chi_{3400}(2729,\cdot)\)
\(\chi_{3400}(2769,\cdot)\)
\(\chi_{3400}(3289,\cdot)\)
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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 ]
\((2551,1701,2177,1601)\) → \((1,1,e\left(\frac{7}{10}\right),e\left(\frac{1}{8}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 3400 }(9, a) \) |
\(1\) | \(1\) | \(e\left(\frac{1}{40}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{3}{40}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{3}{40}\right)\) | \(e\left(\frac{1}{40}\right)\) |
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sage:chi.jacobi_sum(n)
