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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3024, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([18,9,26,6]))
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pari:[g,chi] = znchar(Mod(1739,3024))
| Modulus: | \(3024\) | |
| Conductor: | \(3024\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(36\) |
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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
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| Minimal: | yes |
| Parity: | odd |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
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\(\chi_{3024}(131,\cdot)\)
\(\chi_{3024}(227,\cdot)\)
\(\chi_{3024}(635,\cdot)\)
\(\chi_{3024}(731,\cdot)\)
\(\chi_{3024}(1139,\cdot)\)
\(\chi_{3024}(1235,\cdot)\)
\(\chi_{3024}(1643,\cdot)\)
\(\chi_{3024}(1739,\cdot)\)
\(\chi_{3024}(2147,\cdot)\)
\(\chi_{3024}(2243,\cdot)\)
\(\chi_{3024}(2651,\cdot)\)
\(\chi_{3024}(2747,\cdot)\)
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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 ]
\((1135,757,785,2593)\) → \((-1,i,e\left(\frac{13}{18}\right),e\left(\frac{1}{6}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) |
| \( \chi_{ 3024 }(1739, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{25}{36}\right)\) | \(e\left(\frac{29}{36}\right)\) | \(e\left(\frac{1}{36}\right)\) | \(1\) | \(-i\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{17}{36}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{11}{12}\right)\) |
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sage:chi.jacobi_sum(n)
