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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8450, base_ring=CyclotomicField(780))
M = H._module
chi = DirichletCharacter(H, M([117,355]))
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pari:[g,chi] = znchar(Mod(33,8450))
\(\chi_{8450}(33,\cdot)\)
\(\chi_{8450}(63,\cdot)\)
\(\chi_{8450}(67,\cdot)\)
\(\chi_{8450}(97,\cdot)\)
\(\chi_{8450}(163,\cdot)\)
\(\chi_{8450}(197,\cdot)\)
\(\chi_{8450}(227,\cdot)\)
\(\chi_{8450}(323,\cdot)\)
\(\chi_{8450}(327,\cdot)\)
\(\chi_{8450}(423,\cdot)\)
\(\chi_{8450}(453,\cdot)\)
\(\chi_{8450}(487,\cdot)\)
\(\chi_{8450}(553,\cdot)\)
\(\chi_{8450}(583,\cdot)\)
\(\chi_{8450}(617,\cdot)\)
\(\chi_{8450}(683,\cdot)\)
\(\chi_{8450}(713,\cdot)\)
\(\chi_{8450}(717,\cdot)\)
\(\chi_{8450}(747,\cdot)\)
\(\chi_{8450}(813,\cdot)\)
\(\chi_{8450}(847,\cdot)\)
\(\chi_{8450}(877,\cdot)\)
\(\chi_{8450}(973,\cdot)\)
\(\chi_{8450}(977,\cdot)\)
\(\chi_{8450}(1073,\cdot)\)
\(\chi_{8450}(1137,\cdot)\)
\(\chi_{8450}(1203,\cdot)\)
\(\chi_{8450}(1233,\cdot)\)
\(\chi_{8450}(1237,\cdot)\)
\(\chi_{8450}(1267,\cdot)\)
...
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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 ]
\((677,3551)\) → \((e\left(\frac{3}{20}\right),e\left(\frac{71}{156}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 8450 }(33, a) \) |
\(1\) | \(1\) | \(e\left(\frac{379}{780}\right)\) | \(e\left(\frac{35}{78}\right)\) | \(e\left(\frac{379}{390}\right)\) | \(e\left(\frac{217}{780}\right)\) | \(e\left(\frac{311}{780}\right)\) | \(e\left(\frac{17}{60}\right)\) | \(e\left(\frac{243}{260}\right)\) | \(e\left(\frac{49}{60}\right)\) | \(e\left(\frac{119}{260}\right)\) | \(e\left(\frac{197}{390}\right)\) |
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sage:chi.jacobi_sum(n)
