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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(828, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([0,22,63]))
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pari:[g,chi] = znchar(Mod(589,828))
\(\chi_{828}(61,\cdot)\)
\(\chi_{828}(97,\cdot)\)
\(\chi_{828}(157,\cdot)\)
\(\chi_{828}(205,\cdot)\)
\(\chi_{828}(241,\cdot)\)
\(\chi_{828}(313,\cdot)\)
\(\chi_{828}(337,\cdot)\)
\(\chi_{828}(373,\cdot)\)
\(\chi_{828}(385,\cdot)\)
\(\chi_{828}(421,\cdot)\)
\(\chi_{828}(457,\cdot)\)
\(\chi_{828}(481,\cdot)\)
\(\chi_{828}(493,\cdot)\)
\(\chi_{828}(517,\cdot)\)
\(\chi_{828}(589,\cdot)\)
\(\chi_{828}(661,\cdot)\)
\(\chi_{828}(697,\cdot)\)
\(\chi_{828}(709,\cdot)\)
\(\chi_{828}(733,\cdot)\)
\(\chi_{828}(769,\cdot)\)
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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 ]
\((415,461,649)\) → \((1,e\left(\frac{1}{3}\right),e\left(\frac{21}{22}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 828 }(589, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{41}{66}\right)\) | \(e\left(\frac{31}{66}\right)\) | \(e\left(\frac{61}{66}\right)\) | \(e\left(\frac{1}{33}\right)\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{8}{33}\right)\) | \(e\left(\frac{17}{33}\right)\) | \(e\left(\frac{13}{33}\right)\) | \(e\left(\frac{1}{11}\right)\) |
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sage:chi.jacobi_sum(n)
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sage:chi.gauss_sum(a)
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pari:znchargauss(g,chi,a)
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sage:chi.jacobi_sum(n)
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sage:chi.kloosterman_sum(a,b)
