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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3503, base_ring=CyclotomicField(280))
M = H._module
chi = DirichletCharacter(H, M([168,235]))
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pari:[g,chi] = znchar(Mod(1058,3503))
| Modulus: | \(3503\) | |
| Conductor: | \(3503\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(280\) |
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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)
|
\(\chi_{3503}(126,\cdot)\)
\(\chi_{3503}(163,\cdot)\)
\(\chi_{3503}(190,\cdot)\)
\(\chi_{3503}(252,\cdot)\)
\(\chi_{3503}(287,\cdot)\)
\(\chi_{3503}(314,\cdot)\)
\(\chi_{3503}(326,\cdot)\)
\(\chi_{3503}(380,\cdot)\)
\(\chi_{3503}(411,\cdot)\)
\(\chi_{3503}(504,\cdot)\)
\(\chi_{3503}(529,\cdot)\)
\(\chi_{3503}(543,\cdot)\)
\(\chi_{3503}(574,\cdot)\)
\(\chi_{3503}(591,\cdot)\)
\(\chi_{3503}(628,\cdot)\)
\(\chi_{3503}(653,\cdot)\)
\(\chi_{3503}(667,\cdot)\)
\(\chi_{3503}(729,\cdot)\)
\(\chi_{3503}(760,\cdot)\)
\(\chi_{3503}(822,\cdot)\)
\(\chi_{3503}(841,\cdot)\)
\(\chi_{3503}(853,\cdot)\)
\(\chi_{3503}(915,\cdot)\)
\(\chi_{3503}(965,\cdot)\)
\(\chi_{3503}(1008,\cdot)\)
\(\chi_{3503}(1039,\cdot)\)
\(\chi_{3503}(1058,\cdot)\)
\(\chi_{3503}(1089,\cdot)\)
\(\chi_{3503}(1155,\cdot)\)
\(\chi_{3503}(1180,\cdot)\)
...
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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 ]
\((3165,342)\) → \((e\left(\frac{3}{5}\right),e\left(\frac{47}{56}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 3503 }(1058, a) \) |
\(1\) | \(1\) | \(e\left(\frac{33}{70}\right)\) | \(e\left(\frac{123}{280}\right)\) | \(e\left(\frac{33}{35}\right)\) | \(e\left(\frac{37}{56}\right)\) | \(e\left(\frac{51}{56}\right)\) | \(e\left(\frac{18}{35}\right)\) | \(e\left(\frac{29}{70}\right)\) | \(e\left(\frac{123}{140}\right)\) | \(e\left(\frac{37}{280}\right)\) | \(e\left(\frac{127}{140}\right)\) |
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sage:chi.jacobi_sum(n)
