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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8450, base_ring=CyclotomicField(78))
M = H._module
chi = DirichletCharacter(H, M([0,64]))
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pari:[g,chi] = znchar(Mod(6301,8450))
\(\chi_{8450}(451,\cdot)\)
\(\chi_{8450}(601,\cdot)\)
\(\chi_{8450}(1101,\cdot)\)
\(\chi_{8450}(1251,\cdot)\)
\(\chi_{8450}(1751,\cdot)\)
\(\chi_{8450}(1901,\cdot)\)
\(\chi_{8450}(2401,\cdot)\)
\(\chi_{8450}(2551,\cdot)\)
\(\chi_{8450}(3051,\cdot)\)
\(\chi_{8450}(3201,\cdot)\)
\(\chi_{8450}(3701,\cdot)\)
\(\chi_{8450}(3851,\cdot)\)
\(\chi_{8450}(4351,\cdot)\)
\(\chi_{8450}(4501,\cdot)\)
\(\chi_{8450}(5001,\cdot)\)
\(\chi_{8450}(5151,\cdot)\)
\(\chi_{8450}(5651,\cdot)\)
\(\chi_{8450}(5801,\cdot)\)
\(\chi_{8450}(6301,\cdot)\)
\(\chi_{8450}(6451,\cdot)\)
\(\chi_{8450}(7101,\cdot)\)
\(\chi_{8450}(7601,\cdot)\)
\(\chi_{8450}(8251,\cdot)\)
\(\chi_{8450}(8401,\cdot)\)
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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 ]
\((677,3551)\) → \((1,e\left(\frac{32}{39}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 8450 }(6301, a) \) |
\(1\) | \(1\) | \(e\left(\frac{29}{39}\right)\) | \(e\left(\frac{31}{39}\right)\) | \(e\left(\frac{19}{39}\right)\) | \(e\left(\frac{20}{39}\right)\) | \(e\left(\frac{31}{39}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{32}{39}\right)\) |
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sage:chi.jacobi_sum(n)
