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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3165, base_ring=CyclotomicField(210))
M = H._module
chi = DirichletCharacter(H, M([0,105,104]))
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gp:[g,chi] = znchar(Mod(949, 3165))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("3165.949");
| Modulus: | \(3165\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(1055\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(210\) |
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sage:chi.multiplicative_order()
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gp:charorder(g,chi)
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magma:Order(chi);
|
| Real: | no |
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sage:chi.multiplicative_order() <= 2
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gp:charorder(g,chi) <= 2
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magma:Order(chi) le 2;
|
| Primitive: | no, induced from \(\chi_{1055}(949,\cdot)\) |
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sage:chi.is_primitive()
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gp:#znconreyconductor(g,chi)==1
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magma:IsPrimitive(chi);
|
| Minimal: | yes |
| Parity: | even |
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sage:chi.is_odd()
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gp:zncharisodd(g,chi)
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magma:IsOdd(chi);
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\(\chi_{3165}(4,\cdot)\)
\(\chi_{3165}(49,\cdot)\)
\(\chi_{3165}(139,\cdot)\)
\(\chi_{3165}(154,\cdot)\)
\(\chi_{3165}(289,\cdot)\)
\(\chi_{3165}(304,\cdot)\)
\(\chi_{3165}(469,\cdot)\)
\(\chi_{3165}(484,\cdot)\)
\(\chi_{3165}(604,\cdot)\)
\(\chi_{3165}(649,\cdot)\)
\(\chi_{3165}(679,\cdot)\)
\(\chi_{3165}(769,\cdot)\)
\(\chi_{3165}(874,\cdot)\)
\(\chi_{3165}(889,\cdot)\)
\(\chi_{3165}(949,\cdot)\)
\(\chi_{3165}(964,\cdot)\)
\(\chi_{3165}(1099,\cdot)\)
\(\chi_{3165}(1114,\cdot)\)
\(\chi_{3165}(1174,\cdot)\)
\(\chi_{3165}(1249,\cdot)\)
\(\chi_{3165}(1264,\cdot)\)
\(\chi_{3165}(1369,\cdot)\)
\(\chi_{3165}(1429,\cdot)\)
\(\chi_{3165}(1474,\cdot)\)
\(\chi_{3165}(1744,\cdot)\)
\(\chi_{3165}(1864,\cdot)\)
\(\chi_{3165}(1969,\cdot)\)
\(\chi_{3165}(2119,\cdot)\)
\(\chi_{3165}(2134,\cdot)\)
\(\chi_{3165}(2179,\cdot)\)
...
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sage:chi.galois_orbit()
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gp:order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
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magma:order := Order(chi);
{ chi^k : k in [1..order-1] | GCD(k,order) eq 1 };
\((2111,1267,2956)\) → \((1,-1,e\left(\frac{52}{105}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 3165 }(949, a) \) |
\(1\) | \(1\) | \(e\left(\frac{209}{210}\right)\) | \(e\left(\frac{104}{105}\right)\) | \(e\left(\frac{71}{210}\right)\) | \(e\left(\frac{69}{70}\right)\) | \(e\left(\frac{8}{35}\right)\) | \(e\left(\frac{57}{70}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{103}{105}\right)\) | \(e\left(\frac{11}{210}\right)\) | \(e\left(\frac{4}{15}\right)\) |
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sage:chi(x)
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gp:chareval(g,chi,x) \\\\ x integer, value in Q/Z'
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magma:chi(x)
