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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(24863, base_ring=CyclotomicField(506))
M = H._module
chi = DirichletCharacter(H, M([2,319]))
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gp:[g,chi] = znchar(Mod(2141, 24863))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("24863.2141");
| Modulus: | \(24863\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(24863\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(506\) |
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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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\(\chi_{24863}(154,\cdot)\)
\(\chi_{24863}(372,\cdot)\)
\(\chi_{24863}(443,\cdot)\)
\(\chi_{24863}(445,\cdot)\)
\(\chi_{24863}(593,\cdot)\)
\(\chi_{24863}(749,\cdot)\)
\(\chi_{24863}(790,\cdot)\)
\(\chi_{24863}(876,\cdot)\)
\(\chi_{24863}(933,\cdot)\)
\(\chi_{24863}(1020,\cdot)\)
\(\chi_{24863}(1067,\cdot)\)
\(\chi_{24863}(1143,\cdot)\)
\(\chi_{24863}(1204,\cdot)\)
\(\chi_{24863}(1393,\cdot)\)
\(\chi_{24863}(1497,\cdot)\)
\(\chi_{24863}(1566,\cdot)\)
\(\chi_{24863}(1582,\cdot)\)
\(\chi_{24863}(1637,\cdot)\)
\(\chi_{24863}(1750,\cdot)\)
\(\chi_{24863}(1754,\cdot)\)
\(\chi_{24863}(1846,\cdot)\)
\(\chi_{24863}(1876,\cdot)\)
\(\chi_{24863}(1950,\cdot)\)
\(\chi_{24863}(1967,\cdot)\)
\(\chi_{24863}(1984,\cdot)\)
\(\chi_{24863}(2065,\cdot)\)
\(\chi_{24863}(2141,\cdot)\)
\(\chi_{24863}(2203,\cdot)\)
\(\chi_{24863}(2332,\cdot)\)
\(\chi_{24863}(2428,\cdot)\)
...
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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 };
\((16404,8465)\) → \((e\left(\frac{1}{253}\right),e\left(\frac{29}{46}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 24863 }(2141, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{35}{253}\right)\) | \(e\left(\frac{170}{253}\right)\) | \(e\left(\frac{70}{253}\right)\) | \(e\left(\frac{321}{506}\right)\) | \(e\left(\frac{205}{253}\right)\) | \(e\left(\frac{173}{253}\right)\) | \(e\left(\frac{105}{253}\right)\) | \(e\left(\frac{87}{253}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{95}{506}\right)\) |
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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)
