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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(134505, base_ring=CyclotomicField(420))
M = H._module
chi = DirichletCharacter(H, M([70,315,80,133]))
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gp:[g,chi] = znchar(Mod(5483, 134505))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("134505.5483");
| Modulus: | \(134505\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(134505\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(420\) |
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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: | odd |
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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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\(\chi_{134505}(2,\cdot)\)
\(\chi_{134505}(2048,\cdot)\)
\(\chi_{134505}(2552,\cdot)\)
\(\chi_{134505}(3812,\cdot)\)
\(\chi_{134505}(5483,\cdot)\)
\(\chi_{134505}(7058,\cdot)\)
\(\chi_{134505}(8192,\cdot)\)
\(\chi_{134505}(9608,\cdot)\)
\(\chi_{134505}(9767,\cdot)\)
\(\chi_{134505}(10868,\cdot)\)
\(\chi_{134505}(11372,\cdot)\)
\(\chi_{134505}(15248,\cdot)\)
\(\chi_{134505}(15782,\cdot)\)
\(\chi_{134505}(16853,\cdot)\)
\(\chi_{134505}(19217,\cdot)\)
\(\chi_{134505}(19343,\cdot)\)
\(\chi_{134505}(21263,\cdot)\)
\(\chi_{134505}(21767,\cdot)\)
\(\chi_{134505}(23027,\cdot)\)
\(\chi_{134505}(24698,\cdot)\)
\(\chi_{134505}(26273,\cdot)\)
\(\chi_{134505}(27407,\cdot)\)
\(\chi_{134505}(28823,\cdot)\)
\(\chi_{134505}(28982,\cdot)\)
\(\chi_{134505}(30083,\cdot)\)
\(\chi_{134505}(30587,\cdot)\)
\(\chi_{134505}(34337,\cdot)\)
\(\chi_{134505}(34463,\cdot)\)
\(\chi_{134505}(34997,\cdot)\)
\(\chi_{134505}(36068,\cdot)\)
...
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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 };
\((29891,26902,5491,74971)\) → \((e\left(\frac{1}{6}\right),-i,e\left(\frac{4}{21}\right),e\left(\frac{19}{60}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(8\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) | \(22\) | \(23\) |
| \( \chi_{ 134505 }(5483, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{13}{70}\right)\) | \(e\left(\frac{13}{35}\right)\) | \(e\left(\frac{39}{70}\right)\) | \(e\left(\frac{15}{28}\right)\) | \(e\left(\frac{15}{28}\right)\) | \(e\left(\frac{26}{35}\right)\) | \(e\left(\frac{94}{105}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{101}{140}\right)\) | \(e\left(\frac{13}{35}\right)\) |
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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)
