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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(27209, base_ring=CyclotomicField(858))
M = H._module
chi = DirichletCharacter(H, M([286,154,702]))
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gp:[g,chi] = znchar(Mod(926, 27209))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("27209.926");
| Modulus: | \(27209\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(27209\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(429\) |
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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);
|
\(\chi_{27209}(16,\cdot)\)
\(\chi_{27209}(165,\cdot)\)
\(\chi_{27209}(256,\cdot)\)
\(\chi_{27209}(289,\cdot)\)
\(\chi_{27209}(347,\cdot)\)
\(\chi_{27209}(380,\cdot)\)
\(\chi_{27209}(744,\cdot)\)
\(\chi_{27209}(926,\cdot)\)
\(\chi_{27209}(984,\cdot)\)
\(\chi_{27209}(1108,\cdot)\)
\(\chi_{27209}(1166,\cdot)\)
\(\chi_{27209}(1199,\cdot)\)
\(\chi_{27209}(1290,\cdot)\)
\(\chi_{27209}(1439,\cdot)\)
\(\chi_{27209}(1530,\cdot)\)
\(\chi_{27209}(1803,\cdot)\)
\(\chi_{27209}(1894,\cdot)\)
\(\chi_{27209}(1927,\cdot)\)
\(\chi_{27209}(2076,\cdot)\)
\(\chi_{27209}(2109,\cdot)\)
\(\chi_{27209}(2258,\cdot)\)
\(\chi_{27209}(2349,\cdot)\)
\(\chi_{27209}(2382,\cdot)\)
\(\chi_{27209}(2440,\cdot)\)
\(\chi_{27209}(2473,\cdot)\)
\(\chi_{27209}(2746,\cdot)\)
\(\chi_{27209}(2837,\cdot)\)
\(\chi_{27209}(3077,\cdot)\)
\(\chi_{27209}(3201,\cdot)\)
\(\chi_{27209}(3259,\cdot)\)
...
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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 };
\((3888,3382,5916)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{7}{39}\right),e\left(\frac{9}{11}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
| \( \chi_{ 27209 }(926, a) \) |
\(1\) | \(1\) | \(e\left(\frac{69}{143}\right)\) | \(e\left(\frac{292}{429}\right)\) | \(e\left(\frac{138}{143}\right)\) | \(e\left(\frac{43}{429}\right)\) | \(e\left(\frac{70}{429}\right)\) | \(e\left(\frac{64}{143}\right)\) | \(e\left(\frac{155}{429}\right)\) | \(e\left(\frac{250}{429}\right)\) | \(e\left(\frac{79}{429}\right)\) | \(e\left(\frac{277}{429}\right)\) |
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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)
