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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(361, base_ring=CyclotomicField(342))
M = H._module
chi = DirichletCharacter(H, M([80]))
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pari:[g,chi] = znchar(Mod(142,361))
| Modulus: | \(361\) | |
| Conductor: | \(361\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(171\) |
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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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\(\chi_{361}(4,\cdot)\)
\(\chi_{361}(5,\cdot)\)
\(\chi_{361}(6,\cdot)\)
\(\chi_{361}(9,\cdot)\)
\(\chi_{361}(16,\cdot)\)
\(\chi_{361}(17,\cdot)\)
\(\chi_{361}(23,\cdot)\)
\(\chi_{361}(24,\cdot)\)
\(\chi_{361}(25,\cdot)\)
\(\chi_{361}(35,\cdot)\)
\(\chi_{361}(36,\cdot)\)
\(\chi_{361}(42,\cdot)\)
\(\chi_{361}(43,\cdot)\)
\(\chi_{361}(44,\cdot)\)
\(\chi_{361}(47,\cdot)\)
\(\chi_{361}(55,\cdot)\)
\(\chi_{361}(61,\cdot)\)
\(\chi_{361}(63,\cdot)\)
\(\chi_{361}(66,\cdot)\)
\(\chi_{361}(73,\cdot)\)
\(\chi_{361}(74,\cdot)\)
\(\chi_{361}(80,\cdot)\)
\(\chi_{361}(81,\cdot)\)
\(\chi_{361}(82,\cdot)\)
\(\chi_{361}(85,\cdot)\)
\(\chi_{361}(92,\cdot)\)
\(\chi_{361}(93,\cdot)\)
\(\chi_{361}(100,\cdot)\)
\(\chi_{361}(101,\cdot)\)
\(\chi_{361}(104,\cdot)\)
...
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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 ]
\(2\) → \(e\left(\frac{40}{171}\right)\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 361 }(142, a) \) |
\(1\) | \(1\) | \(e\left(\frac{40}{171}\right)\) | \(e\left(\frac{88}{171}\right)\) | \(e\left(\frac{80}{171}\right)\) | \(e\left(\frac{46}{171}\right)\) | \(e\left(\frac{128}{171}\right)\) | \(e\left(\frac{5}{57}\right)\) | \(e\left(\frac{40}{57}\right)\) | \(e\left(\frac{5}{171}\right)\) | \(e\left(\frac{86}{171}\right)\) | \(e\left(\frac{49}{57}\right)\) |
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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)
