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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(361, base_ring=CyclotomicField(114))
M = H._module
chi = DirichletCharacter(H, M([4]))
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pari:[g,chi] = znchar(Mod(125,361))
| Modulus: | \(361\) | |
| Conductor: | \(361\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(57\) |
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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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\(\chi_{361}(7,\cdot)\)
\(\chi_{361}(11,\cdot)\)
\(\chi_{361}(26,\cdot)\)
\(\chi_{361}(30,\cdot)\)
\(\chi_{361}(45,\cdot)\)
\(\chi_{361}(49,\cdot)\)
\(\chi_{361}(64,\cdot)\)
\(\chi_{361}(83,\cdot)\)
\(\chi_{361}(87,\cdot)\)
\(\chi_{361}(102,\cdot)\)
\(\chi_{361}(106,\cdot)\)
\(\chi_{361}(121,\cdot)\)
\(\chi_{361}(125,\cdot)\)
\(\chi_{361}(140,\cdot)\)
\(\chi_{361}(144,\cdot)\)
\(\chi_{361}(159,\cdot)\)
\(\chi_{361}(163,\cdot)\)
\(\chi_{361}(178,\cdot)\)
\(\chi_{361}(182,\cdot)\)
\(\chi_{361}(197,\cdot)\)
\(\chi_{361}(201,\cdot)\)
\(\chi_{361}(216,\cdot)\)
\(\chi_{361}(220,\cdot)\)
\(\chi_{361}(235,\cdot)\)
\(\chi_{361}(239,\cdot)\)
\(\chi_{361}(254,\cdot)\)
\(\chi_{361}(258,\cdot)\)
\(\chi_{361}(273,\cdot)\)
\(\chi_{361}(277,\cdot)\)
\(\chi_{361}(296,\cdot)\)
...
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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 ]
\(2\) → \(e\left(\frac{2}{57}\right)\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 361 }(125, a) \) |
\(1\) | \(1\) | \(e\left(\frac{2}{57}\right)\) | \(e\left(\frac{50}{57}\right)\) | \(e\left(\frac{4}{57}\right)\) | \(e\left(\frac{8}{57}\right)\) | \(e\left(\frac{52}{57}\right)\) | \(e\left(\frac{5}{19}\right)\) | \(e\left(\frac{2}{19}\right)\) | \(e\left(\frac{43}{57}\right)\) | \(e\left(\frac{10}{57}\right)\) | \(e\left(\frac{11}{19}\right)\) |
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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)
