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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(83, base_ring=CyclotomicField(82))
M = H._module
chi = DirichletCharacter(H, M([35]))
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pari:[g,chi] = znchar(Mod(35,83))
| Modulus: | \(83\) | |
| Conductor: | \(83\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(82\) |
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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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\(\chi_{83}(2,\cdot)\)
\(\chi_{83}(5,\cdot)\)
\(\chi_{83}(6,\cdot)\)
\(\chi_{83}(8,\cdot)\)
\(\chi_{83}(13,\cdot)\)
\(\chi_{83}(14,\cdot)\)
\(\chi_{83}(15,\cdot)\)
\(\chi_{83}(18,\cdot)\)
\(\chi_{83}(19,\cdot)\)
\(\chi_{83}(20,\cdot)\)
\(\chi_{83}(22,\cdot)\)
\(\chi_{83}(24,\cdot)\)
\(\chi_{83}(32,\cdot)\)
\(\chi_{83}(34,\cdot)\)
\(\chi_{83}(35,\cdot)\)
\(\chi_{83}(39,\cdot)\)
\(\chi_{83}(42,\cdot)\)
\(\chi_{83}(43,\cdot)\)
\(\chi_{83}(45,\cdot)\)
\(\chi_{83}(46,\cdot)\)
\(\chi_{83}(47,\cdot)\)
\(\chi_{83}(50,\cdot)\)
\(\chi_{83}(52,\cdot)\)
\(\chi_{83}(53,\cdot)\)
\(\chi_{83}(54,\cdot)\)
\(\chi_{83}(55,\cdot)\)
\(\chi_{83}(56,\cdot)\)
\(\chi_{83}(57,\cdot)\)
\(\chi_{83}(58,\cdot)\)
\(\chi_{83}(60,\cdot)\)
...
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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 ]
\(2\) → \(e\left(\frac{35}{82}\right)\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 83 }(35, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{35}{82}\right)\) | \(e\left(\frac{30}{41}\right)\) | \(e\left(\frac{35}{41}\right)\) | \(e\left(\frac{43}{82}\right)\) | \(e\left(\frac{13}{82}\right)\) | \(e\left(\frac{17}{41}\right)\) | \(e\left(\frac{23}{82}\right)\) | \(e\left(\frac{19}{41}\right)\) | \(e\left(\frac{39}{41}\right)\) | \(e\left(\frac{10}{41}\right)\) |
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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)
