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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(79475, base_ring=CyclotomicField(1360))
M = H._module
chi = DirichletCharacter(H, M([1156,680,695]))
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gp:[g,chi] = znchar(Mod(4597, 79475))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("79475.4597");
| Modulus: | \(79475\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(79475\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(1360\) |
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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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\(\chi_{79475}(142,\cdot)\)
\(\chi_{79475}(197,\cdot)\)
\(\chi_{79475}(428,\cdot)\)
\(\chi_{79475}(813,\cdot)\)
\(\chi_{79475}(912,\cdot)\)
\(\chi_{79475}(923,\cdot)\)
\(\chi_{79475}(1077,\cdot)\)
\(\chi_{79475}(1253,\cdot)\)
\(\chi_{79475}(1363,\cdot)\)
\(\chi_{79475}(1748,\cdot)\)
\(\chi_{79475}(1792,\cdot)\)
\(\chi_{79475}(1847,\cdot)\)
\(\chi_{79475}(1858,\cdot)\)
\(\chi_{79475}(2012,\cdot)\)
\(\chi_{79475}(2067,\cdot)\)
\(\chi_{79475}(2188,\cdot)\)
\(\chi_{79475}(2298,\cdot)\)
\(\chi_{79475}(2683,\cdot)\)
\(\chi_{79475}(2727,\cdot)\)
\(\chi_{79475}(2947,\cdot)\)
\(\chi_{79475}(3002,\cdot)\)
\(\chi_{79475}(3123,\cdot)\)
\(\chi_{79475}(3233,\cdot)\)
\(\chi_{79475}(3662,\cdot)\)
\(\chi_{79475}(3728,\cdot)\)
\(\chi_{79475}(3937,\cdot)\)
\(\chi_{79475}(4058,\cdot)\)
\(\chi_{79475}(4553,\cdot)\)
\(\chi_{79475}(4597,\cdot)\)
\(\chi_{79475}(4652,\cdot)\)
...
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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 };
\((60402,36126,45376)\) → \((e\left(\frac{17}{20}\right),-1,e\left(\frac{139}{272}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(12\) | \(13\) | \(14\) |
| \( \chi_{ 79475 }(4597, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{303}{680}\right)\) | \(e\left(\frac{627}{1360}\right)\) | \(e\left(\frac{303}{340}\right)\) | \(e\left(\frac{1233}{1360}\right)\) | \(e\left(\frac{261}{272}\right)\) | \(e\left(\frac{229}{680}\right)\) | \(e\left(\frac{627}{680}\right)\) | \(e\left(\frac{479}{1360}\right)\) | \(e\left(\frac{69}{85}\right)\) | \(e\left(\frac{551}{1360}\right)\) |
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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)
