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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(201, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([33,4]))
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pari:[g,chi] = znchar(Mod(83,201))
| Modulus: | \(201\) | |
| Conductor: | \(201\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(66\) |
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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
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| Minimal: | yes |
| Parity: | odd |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
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\(\chi_{201}(17,\cdot)\)
\(\chi_{201}(23,\cdot)\)
\(\chi_{201}(26,\cdot)\)
\(\chi_{201}(35,\cdot)\)
\(\chi_{201}(47,\cdot)\)
\(\chi_{201}(56,\cdot)\)
\(\chi_{201}(65,\cdot)\)
\(\chi_{201}(71,\cdot)\)
\(\chi_{201}(77,\cdot)\)
\(\chi_{201}(83,\cdot)\)
\(\chi_{201}(86,\cdot)\)
\(\chi_{201}(116,\cdot)\)
\(\chi_{201}(122,\cdot)\)
\(\chi_{201}(140,\cdot)\)
\(\chi_{201}(155,\cdot)\)
\(\chi_{201}(167,\cdot)\)
\(\chi_{201}(170,\cdot)\)
\(\chi_{201}(173,\cdot)\)
\(\chi_{201}(188,\cdot)\)
\(\chi_{201}(194,\cdot)\)
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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 ]
\((68,136)\) → \((-1,e\left(\frac{2}{33}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 201 }(83, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{37}{66}\right)\) | \(e\left(\frac{4}{33}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{13}{33}\right)\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{32}{33}\right)\) | \(e\left(\frac{5}{66}\right)\) | \(e\left(\frac{5}{33}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{8}{33}\right)\) |
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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)
