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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4851, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([14,1,21]))
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pari:[g,chi] = znchar(Mod(2551,4851))
| Modulus: | \(4851\) | |
| Conductor: | \(4851\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(42\) |
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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
|
| Minimal: | yes |
| Parity: | even |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
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\(\chi_{4851}(439,\cdot)\)
\(\chi_{4851}(1132,\cdot)\)
\(\chi_{4851}(1165,\cdot)\)
\(\chi_{4851}(1825,\cdot)\)
\(\chi_{4851}(1858,\cdot)\)
\(\chi_{4851}(2551,\cdot)\)
\(\chi_{4851}(3211,\cdot)\)
\(\chi_{4851}(3244,\cdot)\)
\(\chi_{4851}(3904,\cdot)\)
\(\chi_{4851}(3937,\cdot)\)
\(\chi_{4851}(4597,\cdot)\)
\(\chi_{4851}(4630,\cdot)\)
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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 ]
\((4313,199,442)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{1}{42}\right),-1)\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(13\) | \(16\) | \(17\) | \(19\) | \(20\) |
| \( \chi_{ 4851 }(2551, a) \) |
\(1\) | \(1\) | \(e\left(\frac{19}{42}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{11}{42}\right)\) |
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sage:chi.jacobi_sum(n)
