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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5001, base_ring=CyclotomicField(34))
M = H._module
chi = DirichletCharacter(H, M([17,31]))
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pari:[g,chi] = znchar(Mod(263,5001))
| Modulus: | \(5001\) | |
| Conductor: | \(5001\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(34\) |
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sage:chi.multiplicative_order()
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pari:charorder(g,chi)
|
| Real: | no |
| Primitive: | yes |
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sage:chi.is_primitive()
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pari:#znconreyconductor(g,chi)==1
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| Minimal: | yes |
| Parity: | even |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
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\(\chi_{5001}(263,\cdot)\)
\(\chi_{5001}(290,\cdot)\)
\(\chi_{5001}(326,\cdot)\)
\(\chi_{5001}(479,\cdot)\)
\(\chi_{5001}(605,\cdot)\)
\(\chi_{5001}(845,\cdot)\)
\(\chi_{5001}(917,\cdot)\)
\(\chi_{5001}(1025,\cdot)\)
\(\chi_{5001}(1118,\cdot)\)
\(\chi_{5001}(2810,\cdot)\)
\(\chi_{5001}(3746,\cdot)\)
\(\chi_{5001}(3878,\cdot)\)
\(\chi_{5001}(4049,\cdot)\)
\(\chi_{5001}(4124,\cdot)\)
\(\chi_{5001}(4280,\cdot)\)
\(\chi_{5001}(4586,\cdot)\)
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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 ]
\((3335,1669)\) → \((-1,e\left(\frac{31}{34}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 5001 }(263, a) \) |
\(1\) | \(1\) | \(e\left(\frac{7}{17}\right)\) | \(e\left(\frac{14}{17}\right)\) | \(e\left(\frac{16}{17}\right)\) | \(e\left(\frac{1}{34}\right)\) | \(e\left(\frac{4}{17}\right)\) | \(e\left(\frac{6}{17}\right)\) | \(e\left(\frac{29}{34}\right)\) | \(1\) | \(e\left(\frac{15}{34}\right)\) | \(e\left(\frac{11}{17}\right)\) |
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sage:chi.jacobi_sum(n)
