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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8664, base_ring=CyclotomicField(38))
M = H._module
chi = DirichletCharacter(H, M([19,0,19,32]))
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pari:[g,chi] = znchar(Mod(191,8664))
\(\chi_{8664}(191,\cdot)\)
\(\chi_{8664}(647,\cdot)\)
\(\chi_{8664}(1103,\cdot)\)
\(\chi_{8664}(1559,\cdot)\)
\(\chi_{8664}(2015,\cdot)\)
\(\chi_{8664}(2471,\cdot)\)
\(\chi_{8664}(2927,\cdot)\)
\(\chi_{8664}(3383,\cdot)\)
\(\chi_{8664}(3839,\cdot)\)
\(\chi_{8664}(4295,\cdot)\)
\(\chi_{8664}(4751,\cdot)\)
\(\chi_{8664}(5207,\cdot)\)
\(\chi_{8664}(5663,\cdot)\)
\(\chi_{8664}(6119,\cdot)\)
\(\chi_{8664}(6575,\cdot)\)
\(\chi_{8664}(7031,\cdot)\)
\(\chi_{8664}(7487,\cdot)\)
\(\chi_{8664}(8399,\cdot)\)
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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 ]
\((2167,4333,5777,8305)\) → \((-1,1,-1,e\left(\frac{16}{19}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 8664 }(191, a) \) |
\(1\) | \(1\) | \(e\left(\frac{33}{38}\right)\) | \(e\left(\frac{31}{38}\right)\) | \(e\left(\frac{17}{19}\right)\) | \(e\left(\frac{1}{19}\right)\) | \(e\left(\frac{31}{38}\right)\) | \(e\left(\frac{18}{19}\right)\) | \(e\left(\frac{14}{19}\right)\) | \(e\left(\frac{31}{38}\right)\) | \(e\left(\frac{25}{38}\right)\) | \(e\left(\frac{13}{19}\right)\) |
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sage:chi.jacobi_sum(n)
