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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(833, base_ring=CyclotomicField(112))
M = H._module
chi = DirichletCharacter(H, M([8,91]))
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pari:[g,chi] = znchar(Mod(811,833))
| Modulus: | \(833\) | |
| Conductor: | \(833\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(112\) |
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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)
|
\(\chi_{833}(6,\cdot)\)
\(\chi_{833}(20,\cdot)\)
\(\chi_{833}(27,\cdot)\)
\(\chi_{833}(41,\cdot)\)
\(\chi_{833}(62,\cdot)\)
\(\chi_{833}(90,\cdot)\)
\(\chi_{833}(125,\cdot)\)
\(\chi_{833}(139,\cdot)\)
\(\chi_{833}(160,\cdot)\)
\(\chi_{833}(167,\cdot)\)
\(\chi_{833}(181,\cdot)\)
\(\chi_{833}(209,\cdot)\)
\(\chi_{833}(216,\cdot)\)
\(\chi_{833}(258,\cdot)\)
\(\chi_{833}(265,\cdot)\)
\(\chi_{833}(279,\cdot)\)
\(\chi_{833}(286,\cdot)\)
\(\chi_{833}(300,\cdot)\)
\(\chi_{833}(328,\cdot)\)
\(\chi_{833}(335,\cdot)\)
\(\chi_{833}(363,\cdot)\)
\(\chi_{833}(377,\cdot)\)
\(\chi_{833}(384,\cdot)\)
\(\chi_{833}(398,\cdot)\)
\(\chi_{833}(405,\cdot)\)
\(\chi_{833}(419,\cdot)\)
\(\chi_{833}(447,\cdot)\)
\(\chi_{833}(454,\cdot)\)
\(\chi_{833}(482,\cdot)\)
\(\chi_{833}(496,\cdot)\)
...
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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 ]
\((52,785)\) → \((e\left(\frac{1}{14}\right),e\left(\frac{13}{16}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
| \( \chi_{ 833 }(811, a) \) |
\(1\) | \(1\) | \(e\left(\frac{13}{56}\right)\) | \(e\left(\frac{99}{112}\right)\) | \(e\left(\frac{13}{28}\right)\) | \(e\left(\frac{15}{112}\right)\) | \(e\left(\frac{13}{112}\right)\) | \(e\left(\frac{39}{56}\right)\) | \(e\left(\frac{43}{56}\right)\) | \(e\left(\frac{41}{112}\right)\) | \(e\left(\frac{61}{112}\right)\) | \(e\left(\frac{39}{112}\right)\) |
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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)
