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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4563, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([34,33]))
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pari:[g,chi] = znchar(Mod(4037,4563))
\(\chi_{4563}(587,\cdot)\)
\(\chi_{4563}(596,\cdot)\)
\(\chi_{4563}(995,\cdot)\)
\(\chi_{4563}(1202,\cdot)\)
\(\chi_{4563}(2108,\cdot)\)
\(\chi_{4563}(2117,\cdot)\)
\(\chi_{4563}(2516,\cdot)\)
\(\chi_{4563}(2723,\cdot)\)
\(\chi_{4563}(3629,\cdot)\)
\(\chi_{4563}(3638,\cdot)\)
\(\chi_{4563}(4037,\cdot)\)
\(\chi_{4563}(4244,\cdot)\)
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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 ]
\((677,3889)\) → \((e\left(\frac{17}{18}\right),e\left(\frac{11}{12}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(14\) | \(16\) | \(17\) |
| \( \chi_{ 4563 }(4037, a) \) |
\(1\) | \(1\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{35}{36}\right)\) | \(e\left(\frac{7}{36}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{25}{36}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(1\) |
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sage:chi.jacobi_sum(n)
