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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(407, base_ring=CyclotomicField(180))
M = H._module
chi = DirichletCharacter(H, M([162,115]))
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pari:[g,chi] = znchar(Mod(116,407))
| Modulus: | \(407\) | |
| Conductor: | \(407\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(180\) |
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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: | even |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
|
\(\chi_{407}(2,\cdot)\)
\(\chi_{407}(13,\cdot)\)
\(\chi_{407}(17,\cdot)\)
\(\chi_{407}(18,\cdot)\)
\(\chi_{407}(19,\cdot)\)
\(\chi_{407}(24,\cdot)\)
\(\chi_{407}(35,\cdot)\)
\(\chi_{407}(39,\cdot)\)
\(\chi_{407}(50,\cdot)\)
\(\chi_{407}(52,\cdot)\)
\(\chi_{407}(57,\cdot)\)
\(\chi_{407}(61,\cdot)\)
\(\chi_{407}(72,\cdot)\)
\(\chi_{407}(79,\cdot)\)
\(\chi_{407}(94,\cdot)\)
\(\chi_{407}(96,\cdot)\)
\(\chi_{407}(106,\cdot)\)
\(\chi_{407}(116,\cdot)\)
\(\chi_{407}(128,\cdot)\)
\(\chi_{407}(129,\cdot)\)
\(\chi_{407}(150,\cdot)\)
\(\chi_{407}(161,\cdot)\)
\(\chi_{407}(167,\cdot)\)
\(\chi_{407}(172,\cdot)\)
\(\chi_{407}(183,\cdot)\)
\(\chi_{407}(200,\cdot)\)
\(\chi_{407}(204,\cdot)\)
\(\chi_{407}(205,\cdot)\)
\(\chi_{407}(217,\cdot)\)
\(\chi_{407}(227,\cdot)\)
...
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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 ]
\((112,298)\) → \((e\left(\frac{9}{10}\right),e\left(\frac{23}{36}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(12\) |
| \( \chi_{ 407 }(116, a) \) |
\(1\) | \(1\) | \(e\left(\frac{97}{180}\right)\) | \(e\left(\frac{73}{90}\right)\) | \(e\left(\frac{7}{90}\right)\) | \(e\left(\frac{53}{180}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{67}{90}\right)\) | \(e\left(\frac{37}{60}\right)\) | \(e\left(\frac{28}{45}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{8}{9}\right)\) |
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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)
