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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(42237, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([24,9,34]))
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pari:[g,chi] = znchar(Mod(19105,42237))
\(\chi_{42237}(2293,\cdot)\)
\(\chi_{42237}(7519,\cdot)\)
\(\chi_{42237}(8419,\cdot)\)
\(\chi_{42237}(8926,\cdot)\)
\(\chi_{42237}(14386,\cdot)\)
\(\chi_{42237}(18166,\cdot)\)
\(\chi_{42237}(18673,\cdot)\)
\(\chi_{42237}(19105,\cdot)\)
\(\chi_{42237}(24133,\cdot)\)
\(\chi_{42237}(28852,\cdot)\)
\(\chi_{42237}(34783,\cdot)\)
\(\chi_{42237}(40009,\cdot)\)
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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 ]
\((32852,38989,12637)\) → \((e\left(\frac{2}{3}\right),i,e\left(\frac{17}{18}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(14\) | \(16\) | \(17\) |
| \( \chi_{ 42237 }(19105, a) \) |
\(1\) | \(1\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{25}{36}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(-i\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{17}{18}\right)\) |
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sage:chi.jacobi_sum(n)
