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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2023, base_ring=CyclotomicField(204))
M = H._module
chi = DirichletCharacter(H, M([136,69]))
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pari:[g,chi] = znchar(Mod(480,2023))
| Modulus: | \(2023\) | |
| Conductor: | \(2023\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(204\) |
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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)
|
\(\chi_{2023}(4,\cdot)\)
\(\chi_{2023}(30,\cdot)\)
\(\chi_{2023}(72,\cdot)\)
\(\chi_{2023}(81,\cdot)\)
\(\chi_{2023}(123,\cdot)\)
\(\chi_{2023}(149,\cdot)\)
\(\chi_{2023}(191,\cdot)\)
\(\chi_{2023}(200,\cdot)\)
\(\chi_{2023}(242,\cdot)\)
\(\chi_{2023}(268,\cdot)\)
\(\chi_{2023}(310,\cdot)\)
\(\chi_{2023}(319,\cdot)\)
\(\chi_{2023}(361,\cdot)\)
\(\chi_{2023}(387,\cdot)\)
\(\chi_{2023}(429,\cdot)\)
\(\chi_{2023}(438,\cdot)\)
\(\chi_{2023}(480,\cdot)\)
\(\chi_{2023}(506,\cdot)\)
\(\chi_{2023}(548,\cdot)\)
\(\chi_{2023}(557,\cdot)\)
\(\chi_{2023}(599,\cdot)\)
\(\chi_{2023}(625,\cdot)\)
\(\chi_{2023}(667,\cdot)\)
\(\chi_{2023}(676,\cdot)\)
\(\chi_{2023}(718,\cdot)\)
\(\chi_{2023}(744,\cdot)\)
\(\chi_{2023}(786,\cdot)\)
\(\chi_{2023}(795,\cdot)\)
\(\chi_{2023}(837,\cdot)\)
\(\chi_{2023}(863,\cdot)\)
...
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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 ]
\((290,1737)\) → \((e\left(\frac{2}{3}\right),e\left(\frac{23}{68}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
| \( \chi_{ 2023 }(480, a) \) |
\(1\) | \(1\) | \(e\left(\frac{61}{102}\right)\) | \(e\left(\frac{1}{204}\right)\) | \(e\left(\frac{10}{51}\right)\) | \(e\left(\frac{161}{204}\right)\) | \(e\left(\frac{41}{68}\right)\) | \(e\left(\frac{27}{34}\right)\) | \(e\left(\frac{1}{102}\right)\) | \(e\left(\frac{79}{204}\right)\) | \(e\left(\frac{91}{204}\right)\) | \(e\left(\frac{41}{204}\right)\) |
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sage:chi.jacobi_sum(n)
