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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(841, base_ring=CyclotomicField(812))
M = H._module
chi = DirichletCharacter(H, M([323]))
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pari:[g,chi] = znchar(Mod(172,841))
| Modulus: | \(841\) | |
| Conductor: | \(841\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(812\) |
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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
|
| Minimal: | yes |
| Parity: | odd |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
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\(\chi_{841}(2,\cdot)\)
\(\chi_{841}(3,\cdot)\)
\(\chi_{841}(8,\cdot)\)
\(\chi_{841}(10,\cdot)\)
\(\chi_{841}(11,\cdot)\)
\(\chi_{841}(15,\cdot)\)
\(\chi_{841}(18,\cdot)\)
\(\chi_{841}(19,\cdot)\)
\(\chi_{841}(21,\cdot)\)
\(\chi_{841}(26,\cdot)\)
\(\chi_{841}(27,\cdot)\)
\(\chi_{841}(31,\cdot)\)
\(\chi_{841}(32,\cdot)\)
\(\chi_{841}(37,\cdot)\)
\(\chi_{841}(39,\cdot)\)
\(\chi_{841}(40,\cdot)\)
\(\chi_{841}(43,\cdot)\)
\(\chi_{841}(44,\cdot)\)
\(\chi_{841}(47,\cdot)\)
\(\chi_{841}(48,\cdot)\)
\(\chi_{841}(50,\cdot)\)
\(\chi_{841}(55,\cdot)\)
\(\chi_{841}(56,\cdot)\)
\(\chi_{841}(61,\cdot)\)
\(\chi_{841}(66,\cdot)\)
\(\chi_{841}(68,\cdot)\)
\(\chi_{841}(69,\cdot)\)
\(\chi_{841}(72,\cdot)\)
\(\chi_{841}(73,\cdot)\)
\(\chi_{841}(76,\cdot)\)
...
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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 ]
\(2\) → \(e\left(\frac{323}{812}\right)\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 841 }(172, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{323}{812}\right)\) | \(e\left(\frac{495}{812}\right)\) | \(e\left(\frac{323}{406}\right)\) | \(e\left(\frac{53}{406}\right)\) | \(e\left(\frac{3}{406}\right)\) | \(e\left(\frac{115}{203}\right)\) | \(e\left(\frac{157}{812}\right)\) | \(e\left(\frac{89}{406}\right)\) | \(e\left(\frac{429}{812}\right)\) | \(e\left(\frac{179}{812}\right)\) |
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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)
