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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1083, base_ring=CyclotomicField(114))
M = H._module
chi = DirichletCharacter(H, M([57,1]))
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pari:[g,chi] = znchar(Mod(8,1083))
| Modulus: | \(1083\) | |
| Conductor: | \(1083\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(114\) |
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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)
|
\(\chi_{1083}(8,\cdot)\)
\(\chi_{1083}(50,\cdot)\)
\(\chi_{1083}(65,\cdot)\)
\(\chi_{1083}(107,\cdot)\)
\(\chi_{1083}(122,\cdot)\)
\(\chi_{1083}(164,\cdot)\)
\(\chi_{1083}(179,\cdot)\)
\(\chi_{1083}(221,\cdot)\)
\(\chi_{1083}(236,\cdot)\)
\(\chi_{1083}(278,\cdot)\)
\(\chi_{1083}(335,\cdot)\)
\(\chi_{1083}(350,\cdot)\)
\(\chi_{1083}(392,\cdot)\)
\(\chi_{1083}(407,\cdot)\)
\(\chi_{1083}(449,\cdot)\)
\(\chi_{1083}(464,\cdot)\)
\(\chi_{1083}(506,\cdot)\)
\(\chi_{1083}(521,\cdot)\)
\(\chi_{1083}(563,\cdot)\)
\(\chi_{1083}(578,\cdot)\)
\(\chi_{1083}(620,\cdot)\)
\(\chi_{1083}(635,\cdot)\)
\(\chi_{1083}(677,\cdot)\)
\(\chi_{1083}(692,\cdot)\)
\(\chi_{1083}(734,\cdot)\)
\(\chi_{1083}(749,\cdot)\)
\(\chi_{1083}(806,\cdot)\)
\(\chi_{1083}(848,\cdot)\)
\(\chi_{1083}(863,\cdot)\)
\(\chi_{1083}(905,\cdot)\)
...
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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 ]
\((362,724)\) → \((-1,e\left(\frac{1}{114}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 1083 }(8, a) \) |
\(1\) | \(1\) | \(e\left(\frac{29}{57}\right)\) | \(e\left(\frac{1}{57}\right)\) | \(e\left(\frac{61}{114}\right)\) | \(e\left(\frac{6}{19}\right)\) | \(e\left(\frac{10}{19}\right)\) | \(e\left(\frac{5}{114}\right)\) | \(e\left(\frac{15}{38}\right)\) | \(e\left(\frac{101}{114}\right)\) | \(e\left(\frac{47}{57}\right)\) | \(e\left(\frac{2}{57}\right)\) |
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sage:chi.jacobi_sum(n)
