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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(497, base_ring=CyclotomicField(70))
M = H._module
chi = DirichletCharacter(H, M([35,38]))
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pari:[g,chi] = znchar(Mod(83,497))
| Modulus: | \(497\) | |
| Conductor: | \(497\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(70\) |
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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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\(\chi_{497}(6,\cdot)\)
\(\chi_{497}(27,\cdot)\)
\(\chi_{497}(83,\cdot)\)
\(\chi_{497}(90,\cdot)\)
\(\chi_{497}(111,\cdot)\)
\(\chi_{497}(146,\cdot)\)
\(\chi_{497}(160,\cdot)\)
\(\chi_{497}(202,\cdot)\)
\(\chi_{497}(216,\cdot)\)
\(\chi_{497}(223,\cdot)\)
\(\chi_{497}(237,\cdot)\)
\(\chi_{497}(251,\cdot)\)
\(\chi_{497}(286,\cdot)\)
\(\chi_{497}(293,\cdot)\)
\(\chi_{497}(300,\cdot)\)
\(\chi_{497}(342,\cdot)\)
\(\chi_{497}(363,\cdot)\)
\(\chi_{497}(370,\cdot)\)
\(\chi_{497}(384,\cdot)\)
\(\chi_{497}(391,\cdot)\)
\(\chi_{497}(398,\cdot)\)
\(\chi_{497}(405,\cdot)\)
\(\chi_{497}(419,\cdot)\)
\(\chi_{497}(475,\cdot)\)
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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 ]
\((143,78)\) → \((-1,e\left(\frac{19}{35}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
| \( \chi_{ 497 }(83, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{9}{35}\right)\) | \(e\left(\frac{43}{70}\right)\) | \(e\left(\frac{18}{35}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{61}{70}\right)\) | \(e\left(\frac{27}{35}\right)\) | \(e\left(\frac{8}{35}\right)\) | \(e\left(\frac{67}{70}\right)\) | \(e\left(\frac{29}{35}\right)\) | \(e\left(\frac{9}{70}\right)\) |
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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)
