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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1576, base_ring=CyclotomicField(98))
M = H._module
chi = DirichletCharacter(H, M([49,0,64]))
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pari:[g,chi] = znchar(Mod(175,1576))
\(\chi_{1576}(23,\cdot)\)
\(\chi_{1576}(63,\cdot)\)
\(\chi_{1576}(135,\cdot)\)
\(\chi_{1576}(175,\cdot)\)
\(\chi_{1576}(231,\cdot)\)
\(\chi_{1576}(239,\cdot)\)
\(\chi_{1576}(287,\cdot)\)
\(\chi_{1576}(351,\cdot)\)
\(\chi_{1576}(423,\cdot)\)
\(\chi_{1576}(431,\cdot)\)
\(\chi_{1576}(447,\cdot)\)
\(\chi_{1576}(455,\cdot)\)
\(\chi_{1576}(479,\cdot)\)
\(\chi_{1576}(495,\cdot)\)
\(\chi_{1576}(527,\cdot)\)
\(\chi_{1576}(607,\cdot)\)
\(\chi_{1576}(615,\cdot)\)
\(\chi_{1576}(631,\cdot)\)
\(\chi_{1576}(679,\cdot)\)
\(\chi_{1576}(839,\cdot)\)
\(\chi_{1576}(847,\cdot)\)
\(\chi_{1576}(959,\cdot)\)
\(\chi_{1576}(975,\cdot)\)
\(\chi_{1576}(1039,\cdot)\)
\(\chi_{1576}(1055,\cdot)\)
\(\chi_{1576}(1127,\cdot)\)
\(\chi_{1576}(1135,\cdot)\)
\(\chi_{1576}(1143,\cdot)\)
\(\chi_{1576}(1167,\cdot)\)
\(\chi_{1576}(1175,\cdot)\)
...
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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 ]
\((1183,789,593)\) → \((-1,1,e\left(\frac{32}{49}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 1576 }(175, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{69}{98}\right)\) | \(e\left(\frac{6}{49}\right)\) | \(e\left(\frac{83}{98}\right)\) | \(e\left(\frac{20}{49}\right)\) | \(e\left(\frac{43}{98}\right)\) | \(e\left(\frac{16}{49}\right)\) | \(e\left(\frac{81}{98}\right)\) | \(e\left(\frac{41}{49}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{27}{49}\right)\) |
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sage:chi.jacobi_sum(n)
