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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2255, base_ring=CyclotomicField(10))
M = H._module
chi = DirichletCharacter(H, M([5,8,8]))
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pari:[g,chi] = znchar(Mod(1554,2255))
| Modulus: | \(2255\) | |
| Conductor: | \(2255\) |
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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)
|
| 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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\(\chi_{2255}(994,\cdot)\)
\(\chi_{2255}(1164,\cdot)\)
\(\chi_{2255}(1554,\cdot)\)
\(\chi_{2255}(1699,\cdot)\)
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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 ]
\((452,1641,826)\) → \((-1,e\left(\frac{4}{5}\right),e\left(\frac{4}{5}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(12\) | \(13\) | \(14\) |
| \( \chi_{ 2255 }(1554, a) \) |
\(1\) | \(1\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(1\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{2}{5}\right)\) |
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sage:chi.jacobi_sum(n)
