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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(85547, base_ring=CyclotomicField(330))
M = H._module
chi = DirichletCharacter(H, M([220,12,132]))
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gp:[g,chi] = znchar(Mod(137, 85547))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("85547.137");
| Modulus: | \(85547\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(85547\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(165\) |
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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);
|
\(\chi_{85547}(137,\cdot)\)
\(\chi_{85547}(499,\cdot)\)
\(\chi_{85547}(592,\cdot)\)
\(\chi_{85547}(1248,\cdot)\)
\(\chi_{85547}(2104,\cdot)\)
\(\chi_{85547}(3215,\cdot)\)
\(\chi_{85547}(7165,\cdot)\)
\(\chi_{85547}(7914,\cdot)\)
\(\chi_{85547}(8276,\cdot)\)
\(\chi_{85547}(8369,\cdot)\)
\(\chi_{85547}(9025,\cdot)\)
\(\chi_{85547}(9480,\cdot)\)
\(\chi_{85547}(9881,\cdot)\)
\(\chi_{85547}(10992,\cdot)\)
\(\chi_{85547}(14942,\cdot)\)
\(\chi_{85547}(15691,\cdot)\)
\(\chi_{85547}(16146,\cdot)\)
\(\chi_{85547}(16802,\cdot)\)
\(\chi_{85547}(17257,\cdot)\)
\(\chi_{85547}(17658,\cdot)\)
\(\chi_{85547}(18769,\cdot)\)
\(\chi_{85547}(22719,\cdot)\)
\(\chi_{85547}(23468,\cdot)\)
\(\chi_{85547}(23830,\cdot)\)
\(\chi_{85547}(23923,\cdot)\)
\(\chi_{85547}(24579,\cdot)\)
\(\chi_{85547}(25034,\cdot)\)
\(\chi_{85547}(25435,\cdot)\)
\(\chi_{85547}(26546,\cdot)\)
\(\chi_{85547}(30496,\cdot)\)
...
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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 };
\((61106,49491,48280)\) → \((e\left(\frac{2}{3}\right),e\left(\frac{2}{55}\right),e\left(\frac{2}{5}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(12\) | \(13\) |
| \( \chi_{ 85547 }(137, a) \) |
\(1\) | \(1\) | \(e\left(\frac{127}{165}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{89}{165}\right)\) | \(e\left(\frac{103}{165}\right)\) | \(e\left(\frac{13}{55}\right)\) | \(e\left(\frac{17}{55}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{13}{33}\right)\) | \(e\left(\frac{1}{165}\right)\) | \(e\left(\frac{4}{55}\right)\) |
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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)
