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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(26299, base_ring=CyclotomicField(408))
M = H._module
chi = DirichletCharacter(H, M([68,340,135]))
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gp:[g,chi] = znchar(Mod(2558, 26299))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("26299.2558");
| Modulus: | \(26299\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(26299\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(408\) |
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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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\(\chi_{26299}(264,\cdot)\)
\(\chi_{26299}(355,\cdot)\)
\(\chi_{26299}(920,\cdot)\)
\(\chi_{26299}(1011,\cdot)\)
\(\chi_{26299}(1284,\cdot)\)
\(\chi_{26299}(1375,\cdot)\)
\(\chi_{26299}(1447,\cdot)\)
\(\chi_{26299}(1538,\cdot)\)
\(\chi_{26299}(1811,\cdot)\)
\(\chi_{26299}(1902,\cdot)\)
\(\chi_{26299}(2558,\cdot)\)
\(\chi_{26299}(2831,\cdot)\)
\(\chi_{26299}(2922,\cdot)\)
\(\chi_{26299}(2994,\cdot)\)
\(\chi_{26299}(3085,\cdot)\)
\(\chi_{26299}(3449,\cdot)\)
\(\chi_{26299}(4014,\cdot)\)
\(\chi_{26299}(4105,\cdot)\)
\(\chi_{26299}(4378,\cdot)\)
\(\chi_{26299}(4541,\cdot)\)
\(\chi_{26299}(4632,\cdot)\)
\(\chi_{26299}(4905,\cdot)\)
\(\chi_{26299}(4996,\cdot)\)
\(\chi_{26299}(5561,\cdot)\)
\(\chi_{26299}(5652,\cdot)\)
\(\chi_{26299}(5925,\cdot)\)
\(\chi_{26299}(6016,\cdot)\)
\(\chi_{26299}(6088,\cdot)\)
\(\chi_{26299}(6452,\cdot)\)
\(\chi_{26299}(6543,\cdot)\)
...
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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 };
\((22543,10116,9829)\) → \((e\left(\frac{1}{6}\right),e\left(\frac{5}{6}\right),e\left(\frac{45}{136}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
| \( \chi_{ 26299 }(2558, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{7}{204}\right)\) | \(e\left(\frac{113}{136}\right)\) | \(e\left(\frac{7}{102}\right)\) | \(e\left(\frac{43}{408}\right)\) | \(e\left(\frac{353}{408}\right)\) | \(e\left(\frac{7}{68}\right)\) | \(e\left(\frac{45}{68}\right)\) | \(e\left(\frac{19}{136}\right)\) | \(e\left(\frac{15}{136}\right)\) | \(e\left(\frac{367}{408}\right)\) |
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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)
