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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2028, base_ring=CyclotomicField(156))
M = H._module
chi = DirichletCharacter(H, M([0,0,47]))
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pari:[g,chi] = znchar(Mod(1177,2028))
\(\chi_{2028}(37,\cdot)\)
\(\chi_{2028}(85,\cdot)\)
\(\chi_{2028}(97,\cdot)\)
\(\chi_{2028}(145,\cdot)\)
\(\chi_{2028}(193,\cdot)\)
\(\chi_{2028}(241,\cdot)\)
\(\chi_{2028}(253,\cdot)\)
\(\chi_{2028}(301,\cdot)\)
\(\chi_{2028}(349,\cdot)\)
\(\chi_{2028}(397,\cdot)\)
\(\chi_{2028}(409,\cdot)\)
\(\chi_{2028}(457,\cdot)\)
\(\chi_{2028}(505,\cdot)\)
\(\chi_{2028}(553,\cdot)\)
\(\chi_{2028}(565,\cdot)\)
\(\chi_{2028}(613,\cdot)\)
\(\chi_{2028}(661,\cdot)\)
\(\chi_{2028}(709,\cdot)\)
\(\chi_{2028}(721,\cdot)\)
\(\chi_{2028}(769,\cdot)\)
\(\chi_{2028}(817,\cdot)\)
\(\chi_{2028}(865,\cdot)\)
\(\chi_{2028}(877,\cdot)\)
\(\chi_{2028}(973,\cdot)\)
\(\chi_{2028}(1021,\cdot)\)
\(\chi_{2028}(1081,\cdot)\)
\(\chi_{2028}(1129,\cdot)\)
\(\chi_{2028}(1177,\cdot)\)
\(\chi_{2028}(1189,\cdot)\)
\(\chi_{2028}(1237,\cdot)\)
...
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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 ]
\((1015,677,1861)\) → \((1,1,e\left(\frac{47}{156}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 2028 }(1177, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{37}{52}\right)\) | \(e\left(\frac{37}{156}\right)\) | \(e\left(\frac{5}{156}\right)\) | \(e\left(\frac{77}{78}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{11}{26}\right)\) | \(e\left(\frac{2}{39}\right)\) | \(e\left(\frac{17}{52}\right)\) | \(e\left(\frac{37}{39}\right)\) |
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sage:chi.jacobi_sum(n)
