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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3800, base_ring=CyclotomicField(10))
M = H._module
chi = DirichletCharacter(H, M([0,5,9,5]))
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pari:[g,chi] = znchar(Mod(2469,3800))
| Modulus: | \(3800\) | |
| Conductor: | \(3800\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(10\) |
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sage:chi.multiplicative_order()
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pari:charorder(g,chi)
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| 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: | odd |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
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\(\chi_{3800}(189,\cdot)\)
\(\chi_{3800}(1709,\cdot)\)
\(\chi_{3800}(2469,\cdot)\)
\(\chi_{3800}(3229,\cdot)\)
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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 ]
\((951,1901,1977,401)\) → \((1,-1,e\left(\frac{9}{10}\right),-1)\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 3800 }(2469, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{3}{10}\right)\) | \(-1\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{4}{5}\right)\) |
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sage:chi.jacobi_sum(n)
