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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(309680, base_ring=CyclotomicField(182))
M = H._module
chi = DirichletCharacter(H, M([91,91,0,26,21]))
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pari:[g,chi] = znchar(Mod(71191,309680))
\(\chi_{309680}(71,\cdot)\)
\(\chi_{309680}(2871,\cdot)\)
\(\chi_{309680}(3991,\cdot)\)
\(\chi_{309680}(6231,\cdot)\)
\(\chi_{309680}(17991,\cdot)\)
\(\chi_{309680}(20791,\cdot)\)
\(\chi_{309680}(28631,\cdot)\)
\(\chi_{309680}(33671,\cdot)\)
\(\chi_{309680}(39831,\cdot)\)
\(\chi_{309680}(43191,\cdot)\)
\(\chi_{309680}(44311,\cdot)\)
\(\chi_{309680}(47111,\cdot)\)
\(\chi_{309680}(48231,\cdot)\)
\(\chi_{309680}(65031,\cdot)\)
\(\chi_{309680}(71191,\cdot)\)
\(\chi_{309680}(72871,\cdot)\)
\(\chi_{309680}(75111,\cdot)\)
\(\chi_{309680}(84071,\cdot)\)
\(\chi_{309680}(87431,\cdot)\)
\(\chi_{309680}(88551,\cdot)\)
\(\chi_{309680}(91351,\cdot)\)
\(\chi_{309680}(92471,\cdot)\)
\(\chi_{309680}(94711,\cdot)\)
\(\chi_{309680}(106471,\cdot)\)
\(\chi_{309680}(115431,\cdot)\)
\(\chi_{309680}(119351,\cdot)\)
\(\chi_{309680}(122151,\cdot)\)
\(\chi_{309680}(128311,\cdot)\)
\(\chi_{309680}(131671,\cdot)\)
\(\chi_{309680}(135591,\cdot)\)
...
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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 ]
\((193551,232261,61937,297041,82321)\) → \((-1,-1,1,e\left(\frac{1}{7}\right),e\left(\frac{3}{26}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(27\) | \(29\) | \(31\) |
| \( \chi_{ 309680 }(71191, a) \) |
\(1\) | \(1\) | \(e\left(\frac{47}{182}\right)\) | \(e\left(\frac{47}{91}\right)\) | \(e\left(\frac{51}{91}\right)\) | \(e\left(\frac{25}{182}\right)\) | \(e\left(\frac{181}{182}\right)\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{141}{182}\right)\) | \(e\left(\frac{31}{91}\right)\) | \(e\left(\frac{25}{26}\right)\) |
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sage:chi.jacobi_sum(n)
