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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(841, base_ring=CyclotomicField(406))
M = H._module
chi = DirichletCharacter(H, M([122]))
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pari:[g,chi] = znchar(Mod(139,841))
| Modulus: | \(841\) | |
| Conductor: | \(841\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(203\) |
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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: | even |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
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\(\chi_{841}(7,\cdot)\)
\(\chi_{841}(16,\cdot)\)
\(\chi_{841}(20,\cdot)\)
\(\chi_{841}(23,\cdot)\)
\(\chi_{841}(24,\cdot)\)
\(\chi_{841}(25,\cdot)\)
\(\chi_{841}(36,\cdot)\)
\(\chi_{841}(45,\cdot)\)
\(\chi_{841}(49,\cdot)\)
\(\chi_{841}(52,\cdot)\)
\(\chi_{841}(53,\cdot)\)
\(\chi_{841}(54,\cdot)\)
\(\chi_{841}(65,\cdot)\)
\(\chi_{841}(74,\cdot)\)
\(\chi_{841}(78,\cdot)\)
\(\chi_{841}(81,\cdot)\)
\(\chi_{841}(82,\cdot)\)
\(\chi_{841}(83,\cdot)\)
\(\chi_{841}(94,\cdot)\)
\(\chi_{841}(103,\cdot)\)
\(\chi_{841}(107,\cdot)\)
\(\chi_{841}(110,\cdot)\)
\(\chi_{841}(111,\cdot)\)
\(\chi_{841}(112,\cdot)\)
\(\chi_{841}(123,\cdot)\)
\(\chi_{841}(132,\cdot)\)
\(\chi_{841}(136,\cdot)\)
\(\chi_{841}(139,\cdot)\)
\(\chi_{841}(140,\cdot)\)
\(\chi_{841}(141,\cdot)\)
...
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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 ]
\(2\) → \(e\left(\frac{61}{203}\right)\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 841 }(139, a) \) |
\(1\) | \(1\) | \(e\left(\frac{61}{203}\right)\) | \(e\left(\frac{74}{203}\right)\) | \(e\left(\frac{122}{203}\right)\) | \(e\left(\frac{152}{203}\right)\) | \(e\left(\frac{135}{203}\right)\) | \(e\left(\frac{200}{203}\right)\) | \(e\left(\frac{183}{203}\right)\) | \(e\left(\frac{148}{203}\right)\) | \(e\left(\frac{10}{203}\right)\) | \(e\left(\frac{69}{203}\right)\) |
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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)
