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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1309, base_ring=CyclotomicField(240))
M = H._module
chi = DirichletCharacter(H, M([200,96,15]))
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gp:[g,chi] = znchar(Mod(1006, 1309))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1309.1006");
| Modulus: | \(1309\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(1309\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(240\) |
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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);
|
\(\chi_{1309}(3,\cdot)\)
\(\chi_{1309}(5,\cdot)\)
\(\chi_{1309}(31,\cdot)\)
\(\chi_{1309}(75,\cdot)\)
\(\chi_{1309}(80,\cdot)\)
\(\chi_{1309}(82,\cdot)\)
\(\chi_{1309}(108,\cdot)\)
\(\chi_{1309}(124,\cdot)\)
\(\chi_{1309}(159,\cdot)\)
\(\chi_{1309}(180,\cdot)\)
\(\chi_{1309}(192,\cdot)\)
\(\chi_{1309}(201,\cdot)\)
\(\chi_{1309}(262,\cdot)\)
\(\chi_{1309}(269,\cdot)\)
\(\chi_{1309}(278,\cdot)\)
\(\chi_{1309}(311,\cdot)\)
\(\chi_{1309}(313,\cdot)\)
\(\chi_{1309}(334,\cdot)\)
\(\chi_{1309}(346,\cdot)\)
\(\chi_{1309}(367,\cdot)\)
\(\chi_{1309}(388,\cdot)\)
\(\chi_{1309}(411,\cdot)\)
\(\chi_{1309}(432,\cdot)\)
\(\chi_{1309}(465,\cdot)\)
\(\chi_{1309}(488,\cdot)\)
\(\chi_{1309}(500,\cdot)\)
\(\chi_{1309}(521,\cdot)\)
\(\chi_{1309}(537,\cdot)\)
\(\chi_{1309}(598,\cdot)\)
\(\chi_{1309}(619,\cdot)\)
...
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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 };
\((1123,596,309)\) → \((e\left(\frac{5}{6}\right),e\left(\frac{2}{5}\right),e\left(\frac{1}{16}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(12\) | \(13\) |
| \( \chi_{ 1309 }(1006, a) \) |
\(1\) | \(1\) | \(e\left(\frac{113}{120}\right)\) | \(e\left(\frac{23}{240}\right)\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{19}{240}\right)\) | \(e\left(\frac{3}{80}\right)\) | \(e\left(\frac{33}{40}\right)\) | \(e\left(\frac{23}{120}\right)\) | \(e\left(\frac{1}{48}\right)\) | \(e\left(\frac{47}{48}\right)\) | \(e\left(\frac{3}{20}\right)\) |
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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)
