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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(207, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([55,27]))
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pari:[g,chi] = znchar(Mod(149,207))
| Modulus: | \(207\) | |
| Conductor: | \(207\) |
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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: | even |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
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\(\chi_{207}(5,\cdot)\)
\(\chi_{207}(11,\cdot)\)
\(\chi_{207}(14,\cdot)\)
\(\chi_{207}(20,\cdot)\)
\(\chi_{207}(38,\cdot)\)
\(\chi_{207}(56,\cdot)\)
\(\chi_{207}(65,\cdot)\)
\(\chi_{207}(74,\cdot)\)
\(\chi_{207}(83,\cdot)\)
\(\chi_{207}(86,\cdot)\)
\(\chi_{207}(113,\cdot)\)
\(\chi_{207}(122,\cdot)\)
\(\chi_{207}(149,\cdot)\)
\(\chi_{207}(155,\cdot)\)
\(\chi_{207}(158,\cdot)\)
\(\chi_{207}(176,\cdot)\)
\(\chi_{207}(182,\cdot)\)
\(\chi_{207}(191,\cdot)\)
\(\chi_{207}(194,\cdot)\)
\(\chi_{207}(203,\cdot)\)
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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 ]
\((47,28)\) → \((e\left(\frac{5}{6}\right),e\left(\frac{9}{22}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 207 }(149, a) \) |
\(1\) | \(1\) | \(e\left(\frac{43}{66}\right)\) | \(e\left(\frac{10}{33}\right)\) | \(e\left(\frac{19}{33}\right)\) | \(e\left(\frac{7}{66}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{17}{33}\right)\) | \(e\left(\frac{13}{33}\right)\) | \(e\left(\frac{25}{33}\right)\) | \(e\left(\frac{20}{33}\right)\) |
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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)
