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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(961, base_ring=CyclotomicField(930))
M = H._module
chi = DirichletCharacter(H, M([256]))
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pari:[g,chi] = znchar(Mod(493,961))
| Modulus: | \(961\) | |
| Conductor: | \(961\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(465\) |
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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)
|
\(\chi_{961}(7,\cdot)\)
\(\chi_{961}(9,\cdot)\)
\(\chi_{961}(10,\cdot)\)
\(\chi_{961}(14,\cdot)\)
\(\chi_{961}(18,\cdot)\)
\(\chi_{961}(19,\cdot)\)
\(\chi_{961}(20,\cdot)\)
\(\chi_{961}(28,\cdot)\)
\(\chi_{961}(38,\cdot)\)
\(\chi_{961}(40,\cdot)\)
\(\chi_{961}(41,\cdot)\)
\(\chi_{961}(45,\cdot)\)
\(\chi_{961}(49,\cdot)\)
\(\chi_{961}(50,\cdot)\)
\(\chi_{961}(51,\cdot)\)
\(\chi_{961}(59,\cdot)\)
\(\chi_{961}(69,\cdot)\)
\(\chi_{961}(71,\cdot)\)
\(\chi_{961}(72,\cdot)\)
\(\chi_{961}(76,\cdot)\)
\(\chi_{961}(80,\cdot)\)
\(\chi_{961}(81,\cdot)\)
\(\chi_{961}(82,\cdot)\)
\(\chi_{961}(90,\cdot)\)
\(\chi_{961}(100,\cdot)\)
\(\chi_{961}(102,\cdot)\)
\(\chi_{961}(103,\cdot)\)
\(\chi_{961}(107,\cdot)\)
\(\chi_{961}(111,\cdot)\)
\(\chi_{961}(112,\cdot)\)
...
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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 ]
\(3\) → \(e\left(\frac{128}{465}\right)\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 961 }(493, a) \) |
\(1\) | \(1\) | \(e\left(\frac{119}{155}\right)\) | \(e\left(\frac{128}{465}\right)\) | \(e\left(\frac{83}{155}\right)\) | \(e\left(\frac{11}{93}\right)\) | \(e\left(\frac{4}{93}\right)\) | \(e\left(\frac{59}{465}\right)\) | \(e\left(\frac{47}{155}\right)\) | \(e\left(\frac{256}{465}\right)\) | \(e\left(\frac{412}{465}\right)\) | \(e\left(\frac{424}{465}\right)\) |
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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)
