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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(17675, base_ring=CyclotomicField(300))
M = H._module
chi = DirichletCharacter(H, M([195,100,213]))
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gp:[g,chi] = znchar(Mod(5567, 17675))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("17675.5567");
| Modulus: | \(17675\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(17675\) |
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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: | even |
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sage:chi.is_odd()
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gp:zncharisodd(g,chi)
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magma:IsOdd(chi);
|
\(\chi_{17675}(338,\cdot)\)
\(\chi_{17675}(422,\cdot)\)
\(\chi_{17675}(842,\cdot)\)
\(\chi_{17675}(1563,\cdot)\)
\(\chi_{17675}(1873,\cdot)\)
\(\chi_{17675}(1978,\cdot)\)
\(\chi_{17675}(2172,\cdot)\)
\(\chi_{17675}(2312,\cdot)\)
\(\chi_{17675}(2398,\cdot)\)
\(\chi_{17675}(2522,\cdot)\)
\(\chi_{17675}(2552,\cdot)\)
\(\chi_{17675}(2937,\cdot)\)
\(\chi_{17675}(3042,\cdot)\)
\(\chi_{17675}(3133,\cdot)\)
\(\chi_{17675}(3777,\cdot)\)
\(\chi_{17675}(3798,\cdot)\)
\(\chi_{17675}(4398,\cdot)\)
\(\chi_{17675}(4442,\cdot)\)
\(\chi_{17675}(4503,\cdot)\)
\(\chi_{17675}(4533,\cdot)\)
\(\chi_{17675}(4638,\cdot)\)
\(\chi_{17675}(4923,\cdot)\)
\(\chi_{17675}(5023,\cdot)\)
\(\chi_{17675}(5077,\cdot)\)
\(\chi_{17675}(5177,\cdot)\)
\(\chi_{17675}(5462,\cdot)\)
\(\chi_{17675}(5567,\cdot)\)
\(\chi_{17675}(5597,\cdot)\)
\(\chi_{17675}(5658,\cdot)\)
\(\chi_{17675}(5702,\cdot)\)
...
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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 };
\((12727,15151,15051)\) → \((e\left(\frac{13}{20}\right),e\left(\frac{1}{3}\right),e\left(\frac{71}{100}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) | \(16\) |
| \( \chi_{ 17675 }(5567, a) \) |
\(1\) | \(1\) | \(e\left(\frac{2}{75}\right)\) | \(e\left(\frac{131}{150}\right)\) | \(e\left(\frac{4}{75}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{2}{25}\right)\) | \(e\left(\frac{56}{75}\right)\) | \(e\left(\frac{289}{300}\right)\) | \(e\left(\frac{139}{150}\right)\) | \(e\left(\frac{21}{100}\right)\) | \(e\left(\frac{8}{75}\right)\) |
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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)
