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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2028, base_ring=CyclotomicField(78))
M = H._module
chi = DirichletCharacter(H, M([39,0,61]))
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pari:[g,chi] = znchar(Mod(43,2028))
\(\chi_{2028}(43,\cdot)\)
\(\chi_{2028}(127,\cdot)\)
\(\chi_{2028}(199,\cdot)\)
\(\chi_{2028}(283,\cdot)\)
\(\chi_{2028}(355,\cdot)\)
\(\chi_{2028}(439,\cdot)\)
\(\chi_{2028}(511,\cdot)\)
\(\chi_{2028}(595,\cdot)\)
\(\chi_{2028}(667,\cdot)\)
\(\chi_{2028}(751,\cdot)\)
\(\chi_{2028}(907,\cdot)\)
\(\chi_{2028}(979,\cdot)\)
\(\chi_{2028}(1063,\cdot)\)
\(\chi_{2028}(1135,\cdot)\)
\(\chi_{2028}(1219,\cdot)\)
\(\chi_{2028}(1291,\cdot)\)
\(\chi_{2028}(1447,\cdot)\)
\(\chi_{2028}(1531,\cdot)\)
\(\chi_{2028}(1603,\cdot)\)
\(\chi_{2028}(1687,\cdot)\)
\(\chi_{2028}(1759,\cdot)\)
\(\chi_{2028}(1843,\cdot)\)
\(\chi_{2028}(1915,\cdot)\)
\(\chi_{2028}(1999,\cdot)\)
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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 ]
\((1015,677,1861)\) → \((-1,1,e\left(\frac{61}{78}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 2028 }(43, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{1}{26}\right)\) | \(e\left(\frac{7}{39}\right)\) | \(e\left(\frac{2}{39}\right)\) | \(e\left(\frac{7}{39}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{13}\right)\) | \(e\left(\frac{11}{39}\right)\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{17}{78}\right)\) |
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sage:chi.jacobi_sum(n)
