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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5225, base_ring=CyclotomicField(180))
M = H._module
chi = DirichletCharacter(H, M([117,108,20]))
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gp:[g,chi] = znchar(Mod(42, 5225))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("5225.42");
| Modulus: | \(5225\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(5225\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(180\) |
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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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\(\chi_{5225}(42,\cdot)\)
\(\chi_{5225}(522,\cdot)\)
\(\chi_{5225}(598,\cdot)\)
\(\chi_{5225}(663,\cdot)\)
\(\chi_{5225}(1213,\cdot)\)
\(\chi_{5225}(1347,\cdot)\)
\(\chi_{5225}(1423,\cdot)\)
\(\chi_{5225}(1488,\cdot)\)
\(\chi_{5225}(1852,\cdot)\)
\(\chi_{5225}(1897,\cdot)\)
\(\chi_{5225}(1973,\cdot)\)
\(\chi_{5225}(2038,\cdot)\)
\(\chi_{5225}(2137,\cdot)\)
\(\chi_{5225}(2172,\cdot)\)
\(\chi_{5225}(2248,\cdot)\)
\(\chi_{5225}(2308,\cdot)\)
\(\chi_{5225}(2403,\cdot)\)
\(\chi_{5225}(2517,\cdot)\)
\(\chi_{5225}(2588,\cdot)\)
\(\chi_{5225}(2677,\cdot)\)
\(\chi_{5225}(2722,\cdot)\)
\(\chi_{5225}(2798,\cdot)\)
\(\chi_{5225}(2962,\cdot)\)
\(\chi_{5225}(3133,\cdot)\)
\(\chi_{5225}(3227,\cdot)\)
\(\chi_{5225}(3228,\cdot)\)
\(\chi_{5225}(3272,\cdot)\)
\(\chi_{5225}(3342,\cdot)\)
\(\chi_{5225}(3348,\cdot)\)
\(\chi_{5225}(3502,\cdot)\)
...
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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 };
\((2927,2851,4676)\) → \((e\left(\frac{13}{20}\right),e\left(\frac{3}{5}\right),e\left(\frac{1}{9}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(12\) | \(13\) | \(14\) |
| \( \chi_{ 5225 }(42, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{13}{36}\right)\) | \(e\left(\frac{143}{180}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{7}{45}\right)\) | \(e\left(\frac{7}{60}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{53}{90}\right)\) | \(e\left(\frac{31}{60}\right)\) | \(e\left(\frac{91}{180}\right)\) | \(e\left(\frac{43}{90}\right)\) |
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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)
