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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(115, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([33,4]))
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pari:[g,chi] = znchar(Mod(48,115))
| Modulus: | \(115\) | |
| Conductor: | \(115\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| 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: | odd |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
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\(\chi_{115}(2,\cdot)\)
\(\chi_{115}(3,\cdot)\)
\(\chi_{115}(8,\cdot)\)
\(\chi_{115}(12,\cdot)\)
\(\chi_{115}(13,\cdot)\)
\(\chi_{115}(18,\cdot)\)
\(\chi_{115}(27,\cdot)\)
\(\chi_{115}(32,\cdot)\)
\(\chi_{115}(48,\cdot)\)
\(\chi_{115}(52,\cdot)\)
\(\chi_{115}(58,\cdot)\)
\(\chi_{115}(62,\cdot)\)
\(\chi_{115}(72,\cdot)\)
\(\chi_{115}(73,\cdot)\)
\(\chi_{115}(77,\cdot)\)
\(\chi_{115}(78,\cdot)\)
\(\chi_{115}(82,\cdot)\)
\(\chi_{115}(87,\cdot)\)
\(\chi_{115}(98,\cdot)\)
\(\chi_{115}(108,\cdot)\)
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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 ]
\((47,51)\) → \((-i,e\left(\frac{1}{11}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
| \( \chi_{ 115 }(48, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{41}{44}\right)\) | \(e\left(\frac{31}{44}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{21}{44}\right)\) | \(e\left(\frac{35}{44}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{25}{44}\right)\) | \(e\left(\frac{23}{44}\right)\) |
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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)
