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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4563, base_ring=CyclotomicField(78))
M = H._module
chi = DirichletCharacter(H, M([26,5]))
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pari:[g,chi] = znchar(Mod(10,4563))
\(\chi_{4563}(10,\cdot)\)
\(\chi_{4563}(667,\cdot)\)
\(\chi_{4563}(712,\cdot)\)
\(\chi_{4563}(1018,\cdot)\)
\(\chi_{4563}(1063,\cdot)\)
\(\chi_{4563}(1369,\cdot)\)
\(\chi_{4563}(1414,\cdot)\)
\(\chi_{4563}(1720,\cdot)\)
\(\chi_{4563}(1765,\cdot)\)
\(\chi_{4563}(2071,\cdot)\)
\(\chi_{4563}(2116,\cdot)\)
\(\chi_{4563}(2422,\cdot)\)
\(\chi_{4563}(2467,\cdot)\)
\(\chi_{4563}(2773,\cdot)\)
\(\chi_{4563}(2818,\cdot)\)
\(\chi_{4563}(3124,\cdot)\)
\(\chi_{4563}(3169,\cdot)\)
\(\chi_{4563}(3475,\cdot)\)
\(\chi_{4563}(3520,\cdot)\)
\(\chi_{4563}(3826,\cdot)\)
\(\chi_{4563}(3871,\cdot)\)
\(\chi_{4563}(4177,\cdot)\)
\(\chi_{4563}(4222,\cdot)\)
\(\chi_{4563}(4528,\cdot)\)
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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 ]
\((677,3889)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{5}{78}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(14\) | \(16\) | \(17\) |
| \( \chi_{ 4563 }(10, a) \) |
\(1\) | \(1\) | \(e\left(\frac{31}{78}\right)\) | \(e\left(\frac{31}{39}\right)\) | \(e\left(\frac{19}{78}\right)\) | \(e\left(\frac{5}{26}\right)\) | \(e\left(\frac{5}{26}\right)\) | \(e\left(\frac{25}{39}\right)\) | \(e\left(\frac{73}{78}\right)\) | \(e\left(\frac{23}{39}\right)\) | \(e\left(\frac{23}{39}\right)\) | \(e\left(\frac{14}{39}\right)\) |
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sage:chi.jacobi_sum(n)
