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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(305760, base_ring=CyclotomicField(168))
M = H._module
chi = DirichletCharacter(H, M([84,147,84,0,8,70]))
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pari:[g,chi] = znchar(Mod(69491,305760))
\(\chi_{305760}(11,\cdot)\)
\(\chi_{305760}(3971,\cdot)\)
\(\chi_{305760}(5891,\cdot)\)
\(\chi_{305760}(8171,\cdot)\)
\(\chi_{305760}(21851,\cdot)\)
\(\chi_{305760}(25811,\cdot)\)
\(\chi_{305760}(27731,\cdot)\)
\(\chi_{305760}(30011,\cdot)\)
\(\chi_{305760}(43691,\cdot)\)
\(\chi_{305760}(47651,\cdot)\)
\(\chi_{305760}(49571,\cdot)\)
\(\chi_{305760}(51851,\cdot)\)
\(\chi_{305760}(69491,\cdot)\)
\(\chi_{305760}(73691,\cdot)\)
\(\chi_{305760}(87371,\cdot)\)
\(\chi_{305760}(91331,\cdot)\)
\(\chi_{305760}(93251,\cdot)\)
\(\chi_{305760}(109211,\cdot)\)
\(\chi_{305760}(115091,\cdot)\)
\(\chi_{305760}(117371,\cdot)\)
\(\chi_{305760}(131051,\cdot)\)
\(\chi_{305760}(135011,\cdot)\)
\(\chi_{305760}(136931,\cdot)\)
\(\chi_{305760}(139211,\cdot)\)
\(\chi_{305760}(152891,\cdot)\)
\(\chi_{305760}(156851,\cdot)\)
\(\chi_{305760}(158771,\cdot)\)
\(\chi_{305760}(161051,\cdot)\)
\(\chi_{305760}(174731,\cdot)\)
\(\chi_{305760}(178691,\cdot)\)
...
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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 ]
\((95551,114661,101921,183457,18721,211681)\) → \((-1,e\left(\frac{7}{8}\right),-1,1,e\left(\frac{1}{21}\right),e\left(\frac{5}{12}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(11\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) | \(47\) |
| \( \chi_{ 305760 }(69491, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{11}{56}\right)\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{19}{84}\right)\) | \(e\left(\frac{109}{168}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{53}{168}\right)\) | \(e\left(\frac{37}{42}\right)\) | \(e\left(\frac{167}{168}\right)\) | \(e\left(\frac{83}{84}\right)\) |
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sage:chi.jacobi_sum(n)
