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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6304, base_ring=CyclotomicField(98))
M = H._module
chi = DirichletCharacter(H, M([49,0,40]))
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pari:[g,chi] = znchar(Mod(4703,6304))
\(\chi_{6304}(63,\cdot)\)
\(\chi_{6304}(287,\cdot)\)
\(\chi_{6304}(351,\cdot)\)
\(\chi_{6304}(447,\cdot)\)
\(\chi_{6304}(479,\cdot)\)
\(\chi_{6304}(607,\cdot)\)
\(\chi_{6304}(959,\cdot)\)
\(\chi_{6304}(1055,\cdot)\)
\(\chi_{6304}(1375,\cdot)\)
\(\chi_{6304}(1407,\cdot)\)
\(\chi_{6304}(1439,\cdot)\)
\(\chi_{6304}(1535,\cdot)\)
\(\chi_{6304}(1567,\cdot)\)
\(\chi_{6304}(1599,\cdot)\)
\(\chi_{6304}(2207,\cdot)\)
\(\chi_{6304}(2719,\cdot)\)
\(\chi_{6304}(2751,\cdot)\)
\(\chi_{6304}(3327,\cdot)\)
\(\chi_{6304}(3391,\cdot)\)
\(\chi_{6304}(3583,\cdot)\)
\(\chi_{6304}(3647,\cdot)\)
\(\chi_{6304}(3679,\cdot)\)
\(\chi_{6304}(3999,\cdot)\)
\(\chi_{6304}(4127,\cdot)\)
\(\chi_{6304}(4191,\cdot)\)
\(\chi_{6304}(4287,\cdot)\)
\(\chi_{6304}(4319,\cdot)\)
\(\chi_{6304}(4383,\cdot)\)
\(\chi_{6304}(4415,\cdot)\)
\(\chi_{6304}(4607,\cdot)\)
...
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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 ]
\((1183,3941,3745)\) → \((-1,1,e\left(\frac{20}{49}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 6304 }(4703, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{37}{98}\right)\) | \(e\left(\frac{16}{49}\right)\) | \(e\left(\frac{9}{98}\right)\) | \(e\left(\frac{37}{49}\right)\) | \(e\left(\frac{33}{98}\right)\) | \(e\left(\frac{10}{49}\right)\) | \(e\left(\frac{69}{98}\right)\) | \(e\left(\frac{44}{49}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{23}{49}\right)\) |
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sage:chi.jacobi_sum(n)
