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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([160,96,195]))
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gp:[g,chi] = znchar(Mod(522, 1309))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1309.522");
| 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: | odd |
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sage:chi.is_odd()
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gp:zncharisodd(g,chi)
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magma:IsOdd(chi);
|
\(\chi_{1309}(37,\cdot)\)
\(\chi_{1309}(58,\cdot)\)
\(\chi_{1309}(114,\cdot)\)
\(\chi_{1309}(130,\cdot)\)
\(\chi_{1309}(158,\cdot)\)
\(\chi_{1309}(163,\cdot)\)
\(\chi_{1309}(207,\cdot)\)
\(\chi_{1309}(214,\cdot)\)
\(\chi_{1309}(235,\cdot)\)
\(\chi_{1309}(284,\cdot)\)
\(\chi_{1309}(312,\cdot)\)
\(\chi_{1309}(317,\cdot)\)
\(\chi_{1309}(333,\cdot)\)
\(\chi_{1309}(345,\cdot)\)
\(\chi_{1309}(368,\cdot)\)
\(\chi_{1309}(394,\cdot)\)
\(\chi_{1309}(401,\cdot)\)
\(\chi_{1309}(422,\cdot)\)
\(\chi_{1309}(445,\cdot)\)
\(\chi_{1309}(466,\cdot)\)
\(\chi_{1309}(471,\cdot)\)
\(\chi_{1309}(487,\cdot)\)
\(\chi_{1309}(499,\cdot)\)
\(\chi_{1309}(515,\cdot)\)
\(\chi_{1309}(520,\cdot)\)
\(\chi_{1309}(522,\cdot)\)
\(\chi_{1309}(555,\cdot)\)
\(\chi_{1309}(564,\cdot)\)
\(\chi_{1309}(592,\cdot)\)
\(\chi_{1309}(632,\cdot)\)
...
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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 };
\((1123,596,309)\) → \((e\left(\frac{2}{3}\right),e\left(\frac{2}{5}\right),e\left(\frac{13}{16}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(12\) | \(13\) |
| \( \chi_{ 1309 }(522, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{13}{120}\right)\) | \(e\left(\frac{163}{240}\right)\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{239}{240}\right)\) | \(e\left(\frac{63}{80}\right)\) | \(e\left(\frac{13}{40}\right)\) | \(e\left(\frac{43}{120}\right)\) | \(e\left(\frac{5}{48}\right)\) | \(e\left(\frac{43}{48}\right)\) | \(e\left(\frac{13}{20}\right)\) |
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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)
