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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([67]))
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pari:[g,chi] = znchar(Mod(41,361))
| Modulus: | \(361\) | |
| Conductor: | \(361\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(342\) |
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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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\(\chi_{361}(2,\cdot)\)
\(\chi_{361}(3,\cdot)\)
\(\chi_{361}(10,\cdot)\)
\(\chi_{361}(13,\cdot)\)
\(\chi_{361}(14,\cdot)\)
\(\chi_{361}(15,\cdot)\)
\(\chi_{361}(21,\cdot)\)
\(\chi_{361}(22,\cdot)\)
\(\chi_{361}(29,\cdot)\)
\(\chi_{361}(32,\cdot)\)
\(\chi_{361}(33,\cdot)\)
\(\chi_{361}(34,\cdot)\)
\(\chi_{361}(40,\cdot)\)
\(\chi_{361}(41,\cdot)\)
\(\chi_{361}(48,\cdot)\)
\(\chi_{361}(51,\cdot)\)
\(\chi_{361}(52,\cdot)\)
\(\chi_{361}(53,\cdot)\)
\(\chi_{361}(59,\cdot)\)
\(\chi_{361}(60,\cdot)\)
\(\chi_{361}(67,\cdot)\)
\(\chi_{361}(70,\cdot)\)
\(\chi_{361}(71,\cdot)\)
\(\chi_{361}(72,\cdot)\)
\(\chi_{361}(78,\cdot)\)
\(\chi_{361}(79,\cdot)\)
\(\chi_{361}(86,\cdot)\)
\(\chi_{361}(89,\cdot)\)
\(\chi_{361}(90,\cdot)\)
\(\chi_{361}(91,\cdot)\)
...
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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 ]
\(2\) → \(e\left(\frac{67}{342}\right)\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 361 }(41, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{67}{342}\right)\) | \(e\left(\frac{79}{342}\right)\) | \(e\left(\frac{67}{171}\right)\) | \(e\left(\frac{77}{171}\right)\) | \(e\left(\frac{73}{171}\right)\) | \(e\left(\frac{22}{57}\right)\) | \(e\left(\frac{67}{114}\right)\) | \(e\left(\frac{79}{171}\right)\) | \(e\left(\frac{221}{342}\right)\) | \(e\left(\frac{56}{57}\right)\) |
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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)
