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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(783, base_ring=CyclotomicField(126))
M = H._module
chi = DirichletCharacter(H, M([49,108]))
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pari:[g,chi] = znchar(Mod(20,783))
| Modulus: | \(783\) | |
| Conductor: | \(783\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(126\) |
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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)
|
\(\chi_{783}(20,\cdot)\)
\(\chi_{783}(23,\cdot)\)
\(\chi_{783}(65,\cdot)\)
\(\chi_{783}(74,\cdot)\)
\(\chi_{783}(83,\cdot)\)
\(\chi_{783}(110,\cdot)\)
\(\chi_{783}(140,\cdot)\)
\(\chi_{783}(194,\cdot)\)
\(\chi_{783}(227,\cdot)\)
\(\chi_{783}(239,\cdot)\)
\(\chi_{783}(248,\cdot)\)
\(\chi_{783}(257,\cdot)\)
\(\chi_{783}(281,\cdot)\)
\(\chi_{783}(284,\cdot)\)
\(\chi_{783}(326,\cdot)\)
\(\chi_{783}(335,\cdot)\)
\(\chi_{783}(344,\cdot)\)
\(\chi_{783}(371,\cdot)\)
\(\chi_{783}(401,\cdot)\)
\(\chi_{783}(455,\cdot)\)
\(\chi_{783}(488,\cdot)\)
\(\chi_{783}(500,\cdot)\)
\(\chi_{783}(509,\cdot)\)
\(\chi_{783}(518,\cdot)\)
\(\chi_{783}(542,\cdot)\)
\(\chi_{783}(545,\cdot)\)
\(\chi_{783}(587,\cdot)\)
\(\chi_{783}(596,\cdot)\)
\(\chi_{783}(605,\cdot)\)
\(\chi_{783}(632,\cdot)\)
...
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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 ]
\((407,379)\) → \((e\left(\frac{7}{18}\right),e\left(\frac{6}{7}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 783 }(20, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{31}{126}\right)\) | \(e\left(\frac{31}{63}\right)\) | \(e\left(\frac{101}{126}\right)\) | \(e\left(\frac{32}{63}\right)\) | \(e\left(\frac{31}{42}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{61}{126}\right)\) | \(e\left(\frac{34}{63}\right)\) | \(e\left(\frac{95}{126}\right)\) | \(e\left(\frac{62}{63}\right)\) |
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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)
