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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1014, base_ring=CyclotomicField(78))
M = H._module
chi = DirichletCharacter(H, M([39,35]))
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pari:[g,chi] = znchar(Mod(101,1014))
\(\chi_{1014}(17,\cdot)\)
\(\chi_{1014}(95,\cdot)\)
\(\chi_{1014}(101,\cdot)\)
\(\chi_{1014}(173,\cdot)\)
\(\chi_{1014}(179,\cdot)\)
\(\chi_{1014}(251,\cdot)\)
\(\chi_{1014}(257,\cdot)\)
\(\chi_{1014}(329,\cdot)\)
\(\chi_{1014}(335,\cdot)\)
\(\chi_{1014}(407,\cdot)\)
\(\chi_{1014}(413,\cdot)\)
\(\chi_{1014}(491,\cdot)\)
\(\chi_{1014}(563,\cdot)\)
\(\chi_{1014}(569,\cdot)\)
\(\chi_{1014}(641,\cdot)\)
\(\chi_{1014}(647,\cdot)\)
\(\chi_{1014}(719,\cdot)\)
\(\chi_{1014}(725,\cdot)\)
\(\chi_{1014}(797,\cdot)\)
\(\chi_{1014}(803,\cdot)\)
\(\chi_{1014}(875,\cdot)\)
\(\chi_{1014}(881,\cdot)\)
\(\chi_{1014}(953,\cdot)\)
\(\chi_{1014}(959,\cdot)\)
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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 ]
\((677,847)\) → \((-1,e\left(\frac{35}{78}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 1014 }(101, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{1}{78}\right)\) | \(e\left(\frac{28}{39}\right)\) | \(e\left(\frac{1}{78}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{1}{13}\right)\) | \(e\left(\frac{35}{78}\right)\) | \(e\left(\frac{11}{26}\right)\) | \(e\left(\frac{43}{78}\right)\) |
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sage:chi.jacobi_sum(n)
