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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(264275, base_ring=CyclotomicField(1860))
M = H._module
chi = DirichletCharacter(H, M([279,372,296]))
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gp:[g,chi] = znchar(Mod(8408, 264275))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("264275.8408");
| Modulus: | \(264275\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(264275\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(1860\) |
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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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\(\chi_{264275}(537,\cdot)\)
\(\chi_{264275}(1378,\cdot)\)
\(\chi_{264275}(1538,\cdot)\)
\(\chi_{264275}(2777,\cdot)\)
\(\chi_{264275}(3853,\cdot)\)
\(\chi_{264275}(4288,\cdot)\)
\(\chi_{264275}(4447,\cdot)\)
\(\chi_{264275}(4823,\cdot)\)
\(\chi_{264275}(5042,\cdot)\)
\(\chi_{264275}(5383,\cdot)\)
\(\chi_{264275}(6312,\cdot)\)
\(\chi_{264275}(6352,\cdot)\)
\(\chi_{264275}(6922,\cdot)\)
\(\chi_{264275}(8067,\cdot)\)
\(\chi_{264275}(8398,\cdot)\)
\(\chi_{264275}(8408,\cdot)\)
\(\chi_{264275}(9062,\cdot)\)
\(\chi_{264275}(9903,\cdot)\)
\(\chi_{264275}(10063,\cdot)\)
\(\chi_{264275}(11302,\cdot)\)
\(\chi_{264275}(12813,\cdot)\)
\(\chi_{264275}(12972,\cdot)\)
\(\chi_{264275}(13348,\cdot)\)
\(\chi_{264275}(13567,\cdot)\)
\(\chi_{264275}(13908,\cdot)\)
\(\chi_{264275}(14837,\cdot)\)
\(\chi_{264275}(14877,\cdot)\)
\(\chi_{264275}(15447,\cdot)\)
\(\chi_{264275}(16592,\cdot)\)
\(\chi_{264275}(16923,\cdot)\)
...
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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 };
\((63427,24026,89376)\) → \((e\left(\frac{3}{20}\right),e\left(\frac{1}{5}\right),e\left(\frac{74}{465}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(12\) | \(13\) | \(14\) |
| \( \chi_{ 264275 }(8408, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{81}{124}\right)\) | \(e\left(\frac{301}{372}\right)\) | \(e\left(\frac{19}{62}\right)\) | \(e\left(\frac{43}{93}\right)\) | \(e\left(\frac{1607}{1860}\right)\) | \(e\left(\frac{119}{124}\right)\) | \(e\left(\frac{115}{186}\right)\) | \(e\left(\frac{43}{372}\right)\) | \(e\left(\frac{1609}{1860}\right)\) | \(e\left(\frac{481}{930}\right)\) |
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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)
