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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(161, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([55,54]))
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pari:[g,chi] = znchar(Mod(75,161))
| Modulus: | \(161\) | |
| Conductor: | \(161\) |
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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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\(\chi_{161}(3,\cdot)\)
\(\chi_{161}(12,\cdot)\)
\(\chi_{161}(26,\cdot)\)
\(\chi_{161}(31,\cdot)\)
\(\chi_{161}(52,\cdot)\)
\(\chi_{161}(54,\cdot)\)
\(\chi_{161}(59,\cdot)\)
\(\chi_{161}(73,\cdot)\)
\(\chi_{161}(75,\cdot)\)
\(\chi_{161}(82,\cdot)\)
\(\chi_{161}(87,\cdot)\)
\(\chi_{161}(94,\cdot)\)
\(\chi_{161}(96,\cdot)\)
\(\chi_{161}(101,\cdot)\)
\(\chi_{161}(108,\cdot)\)
\(\chi_{161}(110,\cdot)\)
\(\chi_{161}(117,\cdot)\)
\(\chi_{161}(124,\cdot)\)
\(\chi_{161}(131,\cdot)\)
\(\chi_{161}(150,\cdot)\)
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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 ]
\((24,120)\) → \((e\left(\frac{5}{6}\right),e\left(\frac{9}{11}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
| \( \chi_{ 161 }(75, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{10}{33}\right)\) | \(e\left(\frac{61}{66}\right)\) | \(e\left(\frac{20}{33}\right)\) | \(e\left(\frac{65}{66}\right)\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{28}{33}\right)\) | \(e\left(\frac{19}{66}\right)\) | \(e\left(\frac{23}{33}\right)\) | \(e\left(\frac{35}{66}\right)\) |
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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)
