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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(507, base_ring=CyclotomicField(52))
M = H._module
chi = DirichletCharacter(H, M([26,33]))
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pari:[g,chi] = znchar(Mod(476,507))
| Modulus: | \(507\) | |
| Conductor: | \(507\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(52\) |
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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)
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\(\chi_{507}(5,\cdot)\)
\(\chi_{507}(8,\cdot)\)
\(\chi_{507}(44,\cdot)\)
\(\chi_{507}(47,\cdot)\)
\(\chi_{507}(83,\cdot)\)
\(\chi_{507}(86,\cdot)\)
\(\chi_{507}(122,\cdot)\)
\(\chi_{507}(125,\cdot)\)
\(\chi_{507}(161,\cdot)\)
\(\chi_{507}(164,\cdot)\)
\(\chi_{507}(200,\cdot)\)
\(\chi_{507}(203,\cdot)\)
\(\chi_{507}(242,\cdot)\)
\(\chi_{507}(278,\cdot)\)
\(\chi_{507}(281,\cdot)\)
\(\chi_{507}(317,\cdot)\)
\(\chi_{507}(320,\cdot)\)
\(\chi_{507}(356,\cdot)\)
\(\chi_{507}(359,\cdot)\)
\(\chi_{507}(395,\cdot)\)
\(\chi_{507}(398,\cdot)\)
\(\chi_{507}(434,\cdot)\)
\(\chi_{507}(473,\cdot)\)
\(\chi_{507}(476,\cdot)\)
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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 ]
\((170,340)\) → \((-1,e\left(\frac{33}{52}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(14\) | \(16\) | \(17\) |
| \( \chi_{ 507 }(476, a) \) |
\(1\) | \(1\) | \(e\left(\frac{7}{52}\right)\) | \(e\left(\frac{7}{26}\right)\) | \(e\left(\frac{11}{52}\right)\) | \(e\left(\frac{47}{52}\right)\) | \(e\left(\frac{21}{52}\right)\) | \(e\left(\frac{9}{26}\right)\) | \(e\left(\frac{45}{52}\right)\) | \(e\left(\frac{1}{26}\right)\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{2}{13}\right)\) |
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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)
