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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(23177, base_ring=CyclotomicField(210))
M = H._module
chi = DirichletCharacter(H, M([75,168,130]))
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gp:[g,chi] = znchar(Mod(531, 23177))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("23177.531");
| Modulus: | \(23177\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(23177\) |
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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: | 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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\(\chi_{23177}(531,\cdot)\)
\(\chi_{23177}(713,\cdot)\)
\(\chi_{23177}(1945,\cdot)\)
\(\chi_{23177}(1973,\cdot)\)
\(\chi_{23177}(2568,\cdot)\)
\(\chi_{23177}(2638,\cdot)\)
\(\chi_{23177}(2820,\cdot)\)
\(\chi_{23177}(3765,\cdot)\)
\(\chi_{23177}(3996,\cdot)\)
\(\chi_{23177}(4052,\cdot)\)
\(\chi_{23177}(4185,\cdot)\)
\(\chi_{23177}(4227,\cdot)\)
\(\chi_{23177}(4745,\cdot)\)
\(\chi_{23177}(5872,\cdot)\)
\(\chi_{23177}(6103,\cdot)\)
\(\chi_{23177}(6187,\cdot)\)
\(\chi_{23177}(6334,\cdot)\)
\(\chi_{23177}(7034,\cdot)\)
\(\chi_{23177}(7979,\cdot)\)
\(\chi_{23177}(8210,\cdot)\)
\(\chi_{23177}(8266,\cdot)\)
\(\chi_{23177}(8441,\cdot)\)
\(\chi_{23177}(8924,\cdot)\)
\(\chi_{23177}(8959,\cdot)\)
\(\chi_{23177}(10079,\cdot)\)
\(\chi_{23177}(11031,\cdot)\)
\(\chi_{23177}(11619,\cdot)\)
\(\chi_{23177}(12186,\cdot)\)
\(\chi_{23177}(12193,\cdot)\)
\(\chi_{23177}(12424,\cdot)\)
...
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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 };
\((21759,4215,16171)\) → \((e\left(\frac{5}{14}\right),e\left(\frac{4}{5}\right),e\left(\frac{13}{21}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(12\) | \(13\) |
| \( \chi_{ 23177 }(531, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{79}{210}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{37}{210}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{79}{105}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{41}{42}\right)\) | \(e\left(\frac{83}{210}\right)\) |
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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)
