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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(805, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([11,22,10]))
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pari:[g,chi] = znchar(Mod(342,805))
| Modulus: | \(805\) | |
| Conductor: | \(805\) |
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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: | odd |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
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\(\chi_{805}(83,\cdot)\)
\(\chi_{805}(97,\cdot)\)
\(\chi_{805}(132,\cdot)\)
\(\chi_{805}(153,\cdot)\)
\(\chi_{805}(237,\cdot)\)
\(\chi_{805}(258,\cdot)\)
\(\chi_{805}(272,\cdot)\)
\(\chi_{805}(293,\cdot)\)
\(\chi_{805}(342,\cdot)\)
\(\chi_{805}(398,\cdot)\)
\(\chi_{805}(412,\cdot)\)
\(\chi_{805}(433,\cdot)\)
\(\chi_{805}(447,\cdot)\)
\(\chi_{805}(503,\cdot)\)
\(\chi_{805}(517,\cdot)\)
\(\chi_{805}(573,\cdot)\)
\(\chi_{805}(608,\cdot)\)
\(\chi_{805}(678,\cdot)\)
\(\chi_{805}(727,\cdot)\)
\(\chi_{805}(797,\cdot)\)
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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 ]
\((162,346,281)\) → \((i,-1,e\left(\frac{5}{22}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) | \(16\) |
| \( \chi_{ 805 }(342, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{31}{44}\right)\) | \(e\left(\frac{39}{44}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{5}{44}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{13}{44}\right)\) | \(e\left(\frac{19}{44}\right)\) | \(e\left(\frac{9}{11}\right)\) |
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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)
