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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3503, base_ring=CyclotomicField(1680))
M = H._module
chi = DirichletCharacter(H, M([56,255]))
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pari:[g,chi] = znchar(Mod(34,3503))
| Modulus: | \(3503\) | |
| Conductor: | \(3503\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(1680\) |
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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)
|
\(\chi_{3503}(3,\cdot)\)
\(\chi_{3503}(12,\cdot)\)
\(\chi_{3503}(17,\cdot)\)
\(\chi_{3503}(21,\cdot)\)
\(\chi_{3503}(24,\cdot)\)
\(\chi_{3503}(34,\cdot)\)
\(\chi_{3503}(43,\cdot)\)
\(\chi_{3503}(55,\cdot)\)
\(\chi_{3503}(74,\cdot)\)
\(\chi_{3503}(75,\cdot)\)
\(\chi_{3503}(79,\cdot)\)
\(\chi_{3503}(84,\cdot)\)
\(\chi_{3503}(86,\cdot)\)
\(\chi_{3503}(96,\cdot)\)
\(\chi_{3503}(110,\cdot)\)
\(\chi_{3503}(136,\cdot)\)
\(\chi_{3503}(137,\cdot)\)
\(\chi_{3503}(146,\cdot)\)
\(\chi_{3503}(158,\cdot)\)
\(\chi_{3503}(167,\cdot)\)
\(\chi_{3503}(168,\cdot)\)
\(\chi_{3503}(172,\cdot)\)
\(\chi_{3503}(179,\cdot)\)
\(\chi_{3503}(189,\cdot)\)
\(\chi_{3503}(197,\cdot)\)
\(\chi_{3503}(199,\cdot)\)
\(\chi_{3503}(203,\cdot)\)
\(\chi_{3503}(207,\cdot)\)
\(\chi_{3503}(220,\cdot)\)
\(\chi_{3503}(229,\cdot)\)
...
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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 ]
\((3165,342)\) → \((e\left(\frac{1}{30}\right),e\left(\frac{17}{112}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 3503 }(34, a) \) |
\(1\) | \(1\) | \(e\left(\frac{87}{140}\right)\) | \(e\left(\frac{311}{1680}\right)\) | \(e\left(\frac{17}{70}\right)\) | \(e\left(\frac{89}{336}\right)\) | \(e\left(\frac{271}{336}\right)\) | \(e\left(\frac{31}{210}\right)\) | \(e\left(\frac{121}{140}\right)\) | \(e\left(\frac{311}{840}\right)\) | \(e\left(\frac{1489}{1680}\right)\) | \(e\left(\frac{839}{840}\right)\) |
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sage:chi.jacobi_sum(n)
