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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([44,45]))
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pari:[g,chi] = znchar(Mod(88,207))
| 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: | odd |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
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\(\chi_{207}(7,\cdot)\)
\(\chi_{207}(34,\cdot)\)
\(\chi_{207}(40,\cdot)\)
\(\chi_{207}(43,\cdot)\)
\(\chi_{207}(61,\cdot)\)
\(\chi_{207}(67,\cdot)\)
\(\chi_{207}(76,\cdot)\)
\(\chi_{207}(79,\cdot)\)
\(\chi_{207}(88,\cdot)\)
\(\chi_{207}(97,\cdot)\)
\(\chi_{207}(103,\cdot)\)
\(\chi_{207}(106,\cdot)\)
\(\chi_{207}(112,\cdot)\)
\(\chi_{207}(130,\cdot)\)
\(\chi_{207}(148,\cdot)\)
\(\chi_{207}(157,\cdot)\)
\(\chi_{207}(166,\cdot)\)
\(\chi_{207}(175,\cdot)\)
\(\chi_{207}(178,\cdot)\)
\(\chi_{207}(205,\cdot)\)
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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 ]
\((47,28)\) → \((e\left(\frac{2}{3}\right),e\left(\frac{15}{22}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 207 }(88, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{1}{33}\right)\) | \(e\left(\frac{2}{33}\right)\) | \(e\left(\frac{1}{66}\right)\) | \(e\left(\frac{41}{66}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{53}{66}\right)\) | \(e\left(\frac{29}{33}\right)\) | \(e\left(\frac{43}{66}\right)\) | \(e\left(\frac{4}{33}\right)\) |
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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)
