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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(220669, base_ring=CyclotomicField(370))
M = H._module
chi = DirichletCharacter(H, M([205,127]))
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gp:[g,chi] = znchar(Mod(34014, 220669))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("220669.34014");
| Modulus: | \(220669\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(220669\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(370\) |
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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);
|
\(\chi_{220669}(1474,\cdot)\)
\(\chi_{220669}(3987,\cdot)\)
\(\chi_{220669}(4077,\cdot)\)
\(\chi_{220669}(4439,\cdot)\)
\(\chi_{220669}(6005,\cdot)\)
\(\chi_{220669}(6135,\cdot)\)
\(\chi_{220669}(11020,\cdot)\)
\(\chi_{220669}(11048,\cdot)\)
\(\chi_{220669}(11874,\cdot)\)
\(\chi_{220669}(12592,\cdot)\)
\(\chi_{220669}(12635,\cdot)\)
\(\chi_{220669}(14241,\cdot)\)
\(\chi_{220669}(15371,\cdot)\)
\(\chi_{220669}(15557,\cdot)\)
\(\chi_{220669}(15615,\cdot)\)
\(\chi_{220669}(16112,\cdot)\)
\(\chi_{220669}(18558,\cdot)\)
\(\chi_{220669}(21673,\cdot)\)
\(\chi_{220669}(22035,\cdot)\)
\(\chi_{220669}(24729,\cdot)\)
\(\chi_{220669}(25013,\cdot)\)
\(\chi_{220669}(25600,\cdot)\)
\(\chi_{220669}(26233,\cdot)\)
\(\chi_{220669}(26930,\cdot)\)
\(\chi_{220669}(27072,\cdot)\)
\(\chi_{220669}(30450,\cdot)\)
\(\chi_{220669}(31205,\cdot)\)
\(\chi_{220669}(34014,\cdot)\)
\(\chi_{220669}(35727,\cdot)\)
\(\chi_{220669}(36961,\cdot)\)
...
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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 };
\((48874,122926)\) → \((e\left(\frac{41}{74}\right),e\left(\frac{127}{370}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 220669 }(34014, a) \) |
\(1\) | \(1\) | \(e\left(\frac{291}{370}\right)\) | \(e\left(\frac{101}{185}\right)\) | \(e\left(\frac{106}{185}\right)\) | \(e\left(\frac{68}{185}\right)\) | \(e\left(\frac{123}{370}\right)\) | \(e\left(\frac{159}{185}\right)\) | \(e\left(\frac{133}{370}\right)\) | \(e\left(\frac{17}{185}\right)\) | \(e\left(\frac{57}{370}\right)\) | \(e\left(\frac{3}{37}\right)\) |
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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)
