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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2784, base_ring=CyclotomicField(8))
M = H._module
chi = DirichletCharacter(H, M([0,3,4,6]))
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pari:[g,chi] = znchar(Mod(1757,2784))
| Modulus: | \(2784\) | |
| Conductor: | \(2784\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(8\) |
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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: | even |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
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\(\chi_{2784}(365,\cdot)\)
\(\chi_{2784}(389,\cdot)\)
\(\chi_{2784}(1757,\cdot)\)
\(\chi_{2784}(1781,\cdot)\)
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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 ]
\((1567,2437,929,1249)\) → \((1,e\left(\frac{3}{8}\right),-1,-i)\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(31\) | \(35\) |
| \( \chi_{ 2784 }(1757, a) \) |
\(1\) | \(1\) | \(e\left(\frac{3}{8}\right)\) | \(-i\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(-i\) | \(e\left(\frac{3}{8}\right)\) | \(-i\) | \(-i\) | \(-i\) | \(e\left(\frac{1}{8}\right)\) |
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sage:chi.jacobi_sum(n)
