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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3150, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([0,27,20]))
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pari:[g,chi] = znchar(Mod(37,3150))
\(\chi_{3150}(37,\cdot)\)
\(\chi_{3150}(163,\cdot)\)
\(\chi_{3150}(487,\cdot)\)
\(\chi_{3150}(613,\cdot)\)
\(\chi_{3150}(667,\cdot)\)
\(\chi_{3150}(1117,\cdot)\)
\(\chi_{3150}(1297,\cdot)\)
\(\chi_{3150}(1423,\cdot)\)
\(\chi_{3150}(1747,\cdot)\)
\(\chi_{3150}(1873,\cdot)\)
\(\chi_{3150}(1927,\cdot)\)
\(\chi_{3150}(2053,\cdot)\)
\(\chi_{3150}(2377,\cdot)\)
\(\chi_{3150}(2503,\cdot)\)
\(\chi_{3150}(2683,\cdot)\)
\(\chi_{3150}(3133,\cdot)\)
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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 ]
\((2801,127,451)\) → \((1,e\left(\frac{9}{20}\right),e\left(\frac{1}{3}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 3150 }(37, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{11}{60}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{37}{60}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{43}{60}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(-i\) |
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sage:chi.jacobi_sum(n)
