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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(46748, base_ring=CyclotomicField(210))
M = H._module
chi = DirichletCharacter(H, M([105,175,135,161]))
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gp:[g,chi] = znchar(Mod(20719, 46748))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("46748.20719");
| Modulus: | \(46748\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(46748\) |
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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: | yes |
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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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\(\chi_{46748}(303,\cdot)\)
\(\chi_{46748}(1687,\cdot)\)
\(\chi_{46748}(2935,\cdot)\)
\(\chi_{46748}(3299,\cdot)\)
\(\chi_{46748}(4703,\cdot)\)
\(\chi_{46748}(5139,\cdot)\)
\(\chi_{46748}(5659,\cdot)\)
\(\chi_{46748}(6211,\cdot)\)
\(\chi_{46748}(6315,\cdot)\)
\(\chi_{46748}(7823,\cdot)\)
\(\chi_{46748}(8883,\cdot)\)
\(\chi_{46748}(9923,\cdot)\)
\(\chi_{46748}(10995,\cdot)\)
\(\chi_{46748}(11535,\cdot)\)
\(\chi_{46748}(12939,\cdot)\)
\(\chi_{46748}(13199,\cdot)\)
\(\chi_{46748}(14219,\cdot)\)
\(\chi_{46748}(14551,\cdot)\)
\(\chi_{46748}(16195,\cdot)\)
\(\chi_{46748}(16423,\cdot)\)
\(\chi_{46748}(19211,\cdot)\)
\(\chi_{46748}(20719,\cdot)\)
\(\chi_{46748}(21031,\cdot)\)
\(\chi_{46748}(24047,\cdot)\)
\(\chi_{46748}(24431,\cdot)\)
\(\chi_{46748}(25003,\cdot)\)
\(\chi_{46748}(25555,\cdot)\)
\(\chi_{46748}(26615,\cdot)\)
\(\chi_{46748}(27447,\cdot)\)
\(\chi_{46748}(29091,\cdot)\)
...
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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 };
\((23375,17981,19345,42225)\) → \((-1,e\left(\frac{5}{6}\right),e\left(\frac{9}{14}\right),e\left(\frac{23}{30}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(15\) | \(17\) | \(19\) | \(21\) | \(23\) |
| \( \chi_{ 46748 }(20719, a) \) |
\(1\) | \(1\) | \(e\left(\frac{57}{70}\right)\) | \(e\left(\frac{41}{42}\right)\) | \(e\left(\frac{89}{105}\right)\) | \(e\left(\frac{22}{35}\right)\) | \(e\left(\frac{4}{105}\right)\) | \(e\left(\frac{83}{105}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{109}{210}\right)\) | \(e\left(\frac{139}{210}\right)\) | \(e\left(\frac{41}{105}\right)\) |
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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)
