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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6319, base_ring=CyclotomicField(770))
M = H._module
chi = DirichletCharacter(H, M([99,735]))
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gp:[g,chi] = znchar(Mod(189, 6319))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("6319.189");
| Modulus: | \(6319\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(6319\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(770\) |
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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);
|
\(\chi_{6319}(11,\cdot)\)
\(\chi_{6319}(22,\cdot)\)
\(\chi_{6319}(44,\cdot)\)
\(\chi_{6319}(133,\cdot)\)
\(\chi_{6319}(139,\cdot)\)
\(\chi_{6319}(170,\cdot)\)
\(\chi_{6319}(189,\cdot)\)
\(\chi_{6319}(203,\cdot)\)
\(\chi_{6319}(235,\cdot)\)
\(\chi_{6319}(265,\cdot)\)
\(\chi_{6319}(278,\cdot)\)
\(\chi_{6319}(317,\cdot)\)
\(\chi_{6319}(340,\cdot)\)
\(\chi_{6319}(352,\cdot)\)
\(\chi_{6319}(437,\cdot)\)
\(\chi_{6319}(470,\cdot)\)
\(\chi_{6319}(489,\cdot)\)
\(\chi_{6319}(495,\cdot)\)
\(\chi_{6319}(518,\cdot)\)
\(\chi_{6319}(530,\cdot)\)
\(\chi_{6319}(532,\cdot)\)
\(\chi_{6319}(556,\cdot)\)
\(\chi_{6319}(559,\cdot)\)
\(\chi_{6319}(615,\cdot)\)
\(\chi_{6319}(621,\cdot)\)
\(\chi_{6319}(667,\cdot)\)
\(\chi_{6319}(704,\cdot)\)
\(\chi_{6319}(708,\cdot)\)
\(\chi_{6319}(723,\cdot)\)
\(\chi_{6319}(762,\cdot)\)
...
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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 };
\((2137,5610)\) → \((e\left(\frac{9}{70}\right),e\left(\frac{21}{22}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 6319 }(189, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{17}{385}\right)\) | \(e\left(\frac{229}{770}\right)\) | \(e\left(\frac{34}{385}\right)\) | \(e\left(\frac{23}{55}\right)\) | \(e\left(\frac{263}{770}\right)\) | \(e\left(\frac{172}{385}\right)\) | \(e\left(\frac{51}{385}\right)\) | \(e\left(\frac{229}{385}\right)\) | \(e\left(\frac{178}{385}\right)\) | \(e\left(\frac{129}{770}\right)\) |
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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)
