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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(107, base_ring=CyclotomicField(106))
M = H._module
chi = DirichletCharacter(H, M([85]))
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pari:[g,chi] = znchar(Mod(78,107))
| Modulus: | \(107\) | |
| Conductor: | \(107\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(106\) |
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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: | odd |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
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\(\chi_{107}(2,\cdot)\)
\(\chi_{107}(5,\cdot)\)
\(\chi_{107}(6,\cdot)\)
\(\chi_{107}(7,\cdot)\)
\(\chi_{107}(8,\cdot)\)
\(\chi_{107}(15,\cdot)\)
\(\chi_{107}(17,\cdot)\)
\(\chi_{107}(18,\cdot)\)
\(\chi_{107}(20,\cdot)\)
\(\chi_{107}(21,\cdot)\)
\(\chi_{107}(22,\cdot)\)
\(\chi_{107}(24,\cdot)\)
\(\chi_{107}(26,\cdot)\)
\(\chi_{107}(28,\cdot)\)
\(\chi_{107}(31,\cdot)\)
\(\chi_{107}(32,\cdot)\)
\(\chi_{107}(38,\cdot)\)
\(\chi_{107}(43,\cdot)\)
\(\chi_{107}(45,\cdot)\)
\(\chi_{107}(46,\cdot)\)
\(\chi_{107}(50,\cdot)\)
\(\chi_{107}(51,\cdot)\)
\(\chi_{107}(54,\cdot)\)
\(\chi_{107}(55,\cdot)\)
\(\chi_{107}(58,\cdot)\)
\(\chi_{107}(59,\cdot)\)
\(\chi_{107}(60,\cdot)\)
\(\chi_{107}(63,\cdot)\)
\(\chi_{107}(65,\cdot)\)
\(\chi_{107}(66,\cdot)\)
...
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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 ]
\(2\) → \(e\left(\frac{85}{106}\right)\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 107 }(78, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{85}{106}\right)\) | \(e\left(\frac{7}{53}\right)\) | \(e\left(\frac{32}{53}\right)\) | \(e\left(\frac{73}{106}\right)\) | \(e\left(\frac{99}{106}\right)\) | \(e\left(\frac{51}{106}\right)\) | \(e\left(\frac{43}{106}\right)\) | \(e\left(\frac{14}{53}\right)\) | \(e\left(\frac{26}{53}\right)\) | \(e\left(\frac{34}{53}\right)\) |
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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)
