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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(125341, base_ring=CyclotomicField(300))
M = H._module
chi = DirichletCharacter(H, M([150,250,213]))
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gp:[g,chi] = znchar(Mod(63137, 125341))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("125341.63137");
| Modulus: | \(125341\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(125341\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(300\) |
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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);
|
\(\chi_{125341}(1104,\cdot)\)
\(\chi_{125341}(3569,\cdot)\)
\(\chi_{125341}(3586,\cdot)\)
\(\chi_{125341}(4810,\cdot)\)
\(\chi_{125341}(6068,\cdot)\)
\(\chi_{125341}(8550,\cdot)\)
\(\chi_{125341}(12256,\cdot)\)
\(\chi_{125341}(14738,\cdot)\)
\(\chi_{125341}(15996,\cdot)\)
\(\chi_{125341}(17220,\cdot)\)
\(\chi_{125341}(17237,\cdot)\)
\(\chi_{125341}(19702,\cdot)\)
\(\chi_{125341}(20960,\cdot)\)
\(\chi_{125341}(23425,\cdot)\)
\(\chi_{125341}(25907,\cdot)\)
\(\chi_{125341}(28389,\cdot)\)
\(\chi_{125341}(30871,\cdot)\)
\(\chi_{125341}(30888,\cdot)\)
\(\chi_{125341}(32129,\cdot)\)
\(\chi_{125341}(33370,\cdot)\)
\(\chi_{125341}(35852,\cdot)\)
\(\chi_{125341}(37093,\cdot)\)
\(\chi_{125341}(38317,\cdot)\)
\(\chi_{125341}(38334,\cdot)\)
\(\chi_{125341}(39558,\cdot)\)
\(\chi_{125341}(40816,\cdot)\)
\(\chi_{125341}(43281,\cdot)\)
\(\chi_{125341}(44539,\cdot)\)
\(\chi_{125341}(45780,\cdot)\)
\(\chi_{125341}(50744,\cdot)\)
...
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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 };
\((22120,46360,8688)\) → \((-1,e\left(\frac{5}{6}\right),e\left(\frac{71}{100}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 125341 }(63137, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{113}{300}\right)\) | \(e\left(\frac{49}{100}\right)\) | \(e\left(\frac{113}{150}\right)\) | \(e\left(\frac{28}{75}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{39}{100}\right)\) | \(e\left(\frac{13}{100}\right)\) | \(e\left(\frac{49}{50}\right)\) | \(-i\) | \(e\left(\frac{169}{300}\right)\) |
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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)
