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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([80,168,165]))
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gp:[g,chi] = znchar(Mod(1129, 1309))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1309.1129");
| 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);
|
\(\chi_{1309}(39,\cdot)\)
\(\chi_{1309}(46,\cdot)\)
\(\chi_{1309}(74,\cdot)\)
\(\chi_{1309}(79,\cdot)\)
\(\chi_{1309}(95,\cdot)\)
\(\chi_{1309}(107,\cdot)\)
\(\chi_{1309}(116,\cdot)\)
\(\chi_{1309}(156,\cdot)\)
\(\chi_{1309}(184,\cdot)\)
\(\chi_{1309}(193,\cdot)\)
\(\chi_{1309}(226,\cdot)\)
\(\chi_{1309}(228,\cdot)\)
\(\chi_{1309}(233,\cdot)\)
\(\chi_{1309}(249,\cdot)\)
\(\chi_{1309}(261,\cdot)\)
\(\chi_{1309}(277,\cdot)\)
\(\chi_{1309}(282,\cdot)\)
\(\chi_{1309}(303,\cdot)\)
\(\chi_{1309}(326,\cdot)\)
\(\chi_{1309}(347,\cdot)\)
\(\chi_{1309}(354,\cdot)\)
\(\chi_{1309}(380,\cdot)\)
\(\chi_{1309}(403,\cdot)\)
\(\chi_{1309}(415,\cdot)\)
\(\chi_{1309}(431,\cdot)\)
\(\chi_{1309}(436,\cdot)\)
\(\chi_{1309}(464,\cdot)\)
\(\chi_{1309}(513,\cdot)\)
\(\chi_{1309}(534,\cdot)\)
\(\chi_{1309}(541,\cdot)\)
...
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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 };
\((1123,596,309)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{7}{10}\right),e\left(\frac{11}{16}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(12\) | \(13\) |
| \( \chi_{ 1309 }(1129, a) \) |
\(1\) | \(1\) | \(e\left(\frac{119}{120}\right)\) | \(e\left(\frac{149}{240}\right)\) | \(e\left(\frac{59}{60}\right)\) | \(e\left(\frac{217}{240}\right)\) | \(e\left(\frac{49}{80}\right)\) | \(e\left(\frac{39}{40}\right)\) | \(e\left(\frac{29}{120}\right)\) | \(e\left(\frac{43}{48}\right)\) | \(e\left(\frac{29}{48}\right)\) | \(e\left(\frac{9}{20}\right)\) |
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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)
