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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([39,0,56]))
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pari:[g,chi] = znchar(Mod(55,2028))
\(\chi_{2028}(55,\cdot)\)
\(\chi_{2028}(139,\cdot)\)
\(\chi_{2028}(211,\cdot)\)
\(\chi_{2028}(295,\cdot)\)
\(\chi_{2028}(367,\cdot)\)
\(\chi_{2028}(451,\cdot)\)
\(\chi_{2028}(523,\cdot)\)
\(\chi_{2028}(607,\cdot)\)
\(\chi_{2028}(679,\cdot)\)
\(\chi_{2028}(763,\cdot)\)
\(\chi_{2028}(835,\cdot)\)
\(\chi_{2028}(919,\cdot)\)
\(\chi_{2028}(1075,\cdot)\)
\(\chi_{2028}(1147,\cdot)\)
\(\chi_{2028}(1231,\cdot)\)
\(\chi_{2028}(1303,\cdot)\)
\(\chi_{2028}(1387,\cdot)\)
\(\chi_{2028}(1459,\cdot)\)
\(\chi_{2028}(1615,\cdot)\)
\(\chi_{2028}(1699,\cdot)\)
\(\chi_{2028}(1771,\cdot)\)
\(\chi_{2028}(1855,\cdot)\)
\(\chi_{2028}(1927,\cdot)\)
\(\chi_{2028}(2011,\cdot)\)
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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 ]
\((1015,677,1861)\) → \((-1,1,e\left(\frac{28}{39}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 2028 }(55, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{6}{13}\right)\) | \(e\left(\frac{25}{78}\right)\) | \(e\left(\frac{35}{78}\right)\) | \(e\left(\frac{32}{39}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{28}{39}\right)\) | \(e\left(\frac{15}{26}\right)\) | \(e\left(\frac{61}{78}\right)\) |
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sage:chi.jacobi_sum(n)
