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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(605, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([33,14]))
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pari:[g,chi] = znchar(Mod(593,605))
| Modulus: | \(605\) | |
| Conductor: | \(605\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
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| Order: | \(44\) |
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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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\(\chi_{605}(32,\cdot)\)
\(\chi_{605}(43,\cdot)\)
\(\chi_{605}(87,\cdot)\)
\(\chi_{605}(98,\cdot)\)
\(\chi_{605}(142,\cdot)\)
\(\chi_{605}(153,\cdot)\)
\(\chi_{605}(197,\cdot)\)
\(\chi_{605}(208,\cdot)\)
\(\chi_{605}(252,\cdot)\)
\(\chi_{605}(263,\cdot)\)
\(\chi_{605}(307,\cdot)\)
\(\chi_{605}(318,\cdot)\)
\(\chi_{605}(373,\cdot)\)
\(\chi_{605}(417,\cdot)\)
\(\chi_{605}(428,\cdot)\)
\(\chi_{605}(472,\cdot)\)
\(\chi_{605}(527,\cdot)\)
\(\chi_{605}(538,\cdot)\)
\(\chi_{605}(582,\cdot)\)
\(\chi_{605}(593,\cdot)\)
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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 ]
\((122,486)\) → \((-i,e\left(\frac{7}{22}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(12\) | \(13\) | \(14\) |
| \( \chi_{ 605 }(593, a) \) |
\(1\) | \(1\) | \(e\left(\frac{3}{44}\right)\) | \(i\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{43}{44}\right)\) | \(e\left(\frac{9}{44}\right)\) | \(-1\) | \(e\left(\frac{17}{44}\right)\) | \(e\left(\frac{17}{44}\right)\) | \(e\left(\frac{1}{22}\right)\) |
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sage:chi.jacobi_sum(n)
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sage:chi.gauss_sum(a)
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pari:znchargauss(g,chi,a)
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sage:chi.jacobi_sum(n)
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sage:chi.kloosterman_sum(a,b)
