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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3724, base_ring=CyclotomicField(126))
M = H._module
chi = DirichletCharacter(H, M([63,75,56]))
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pari:[g,chi] = znchar(Mod(2859,3724))
| Modulus: | \(3724\) | |
| Conductor: | \(3724\) |
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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: | even |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
|
\(\chi_{3724}(131,\cdot)\)
\(\chi_{3724}(187,\cdot)\)
\(\chi_{3724}(199,\cdot)\)
\(\chi_{3724}(271,\cdot)\)
\(\chi_{3724}(367,\cdot)\)
\(\chi_{3724}(479,\cdot)\)
\(\chi_{3724}(663,\cdot)\)
\(\chi_{3724}(719,\cdot)\)
\(\chi_{3724}(731,\cdot)\)
\(\chi_{3724}(899,\cdot)\)
\(\chi_{3724}(1251,\cdot)\)
\(\chi_{3724}(1263,\cdot)\)
\(\chi_{3724}(1335,\cdot)\)
\(\chi_{3724}(1431,\cdot)\)
\(\chi_{3724}(1543,\cdot)\)
\(\chi_{3724}(1727,\cdot)\)
\(\chi_{3724}(1867,\cdot)\)
\(\chi_{3724}(1963,\cdot)\)
\(\chi_{3724}(2075,\cdot)\)
\(\chi_{3724}(2259,\cdot)\)
\(\chi_{3724}(2315,\cdot)\)
\(\chi_{3724}(2327,\cdot)\)
\(\chi_{3724}(2399,\cdot)\)
\(\chi_{3724}(2495,\cdot)\)
\(\chi_{3724}(2607,\cdot)\)
\(\chi_{3724}(2791,\cdot)\)
\(\chi_{3724}(2847,\cdot)\)
\(\chi_{3724}(2859,\cdot)\)
\(\chi_{3724}(2931,\cdot)\)
\(\chi_{3724}(3027,\cdot)\)
...
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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 ]
\((1863,3041,3137)\) → \((-1,e\left(\frac{25}{42}\right),e\left(\frac{4}{9}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(23\) | \(25\) | \(27\) |
| \( \chi_{ 3724 }(2859, a) \) |
\(1\) | \(1\) | \(e\left(\frac{55}{63}\right)\) | \(e\left(\frac{47}{126}\right)\) | \(e\left(\frac{47}{63}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{109}{126}\right)\) | \(e\left(\frac{31}{126}\right)\) | \(e\left(\frac{41}{126}\right)\) | \(e\left(\frac{1}{126}\right)\) | \(e\left(\frac{47}{63}\right)\) | \(e\left(\frac{13}{21}\right)\) |
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sage:chi.jacobi_sum(n)
