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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2443, base_ring=CyclotomicField(174))
M = H._module
chi = DirichletCharacter(H, M([116,39]))
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gp:[g,chi] = znchar(Mod(725, 2443))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2443.725");
| Modulus: | \(2443\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(2443\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(174\) |
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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);
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\(\chi_{2443}(37,\cdot)\)
\(\chi_{2443}(60,\cdot)\)
\(\chi_{2443}(86,\cdot)\)
\(\chi_{2443}(100,\cdot)\)
\(\chi_{2443}(121,\cdot)\)
\(\chi_{2443}(261,\cdot)\)
\(\chi_{2443}(282,\cdot)\)
\(\chi_{2443}(366,\cdot)\)
\(\chi_{2443}(394,\cdot)\)
\(\chi_{2443}(424,\cdot)\)
\(\chi_{2443}(429,\cdot)\)
\(\chi_{2443}(464,\cdot)\)
\(\chi_{2443}(527,\cdot)\)
\(\chi_{2443}(632,\cdot)\)
\(\chi_{2443}(667,\cdot)\)
\(\chi_{2443}(725,\cdot)\)
\(\chi_{2443}(746,\cdot)\)
\(\chi_{2443}(758,\cdot)\)
\(\chi_{2443}(767,\cdot)\)
\(\chi_{2443}(823,\cdot)\)
\(\chi_{2443}(837,\cdot)\)
\(\chi_{2443}(879,\cdot)\)
\(\chi_{2443}(921,\cdot)\)
\(\chi_{2443}(1122,\cdot)\)
\(\chi_{2443}(1278,\cdot)\)
\(\chi_{2443}(1355,\cdot)\)
\(\chi_{2443}(1423,\cdot)\)
\(\chi_{2443}(1432,\cdot)\)
\(\chi_{2443}(1444,\cdot)\)
\(\chi_{2443}(1460,\cdot)\)
...
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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 };
\((1746,351)\) → \((e\left(\frac{2}{3}\right),e\left(\frac{13}{58}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
| \( \chi_{ 2443 }(725, a) \) |
\(1\) | \(1\) | \(e\left(\frac{97}{174}\right)\) | \(e\left(\frac{43}{87}\right)\) | \(e\left(\frac{10}{87}\right)\) | \(e\left(\frac{62}{87}\right)\) | \(e\left(\frac{3}{58}\right)\) | \(e\left(\frac{39}{58}\right)\) | \(e\left(\frac{86}{87}\right)\) | \(e\left(\frac{47}{174}\right)\) | \(e\left(\frac{65}{174}\right)\) | \(e\left(\frac{53}{87}\right)\) |
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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)
