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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(152269, base_ring=CyclotomicField(624))
M = H._module
chi = DirichletCharacter(H, M([328,39,444]))
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gp:[g,chi] = znchar(Mod(14011, 152269))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("152269.14011");
| Modulus: | \(152269\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(152269\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(624\) |
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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);
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\(\chi_{152269}(75,\cdot)\)
\(\chi_{152269}(283,\cdot)\)
\(\chi_{152269}(836,\cdot)\)
\(\chi_{152269}(940,\cdot)\)
\(\chi_{152269}(1570,\cdot)\)
\(\chi_{152269}(1876,\cdot)\)
\(\chi_{152269}(2623,\cdot)\)
\(\chi_{152269}(3033,\cdot)\)
\(\chi_{152269}(4261,\cdot)\)
\(\chi_{152269}(4391,\cdot)\)
\(\chi_{152269}(5054,\cdot)\)
\(\chi_{152269}(5327,\cdot)\)
\(\chi_{152269}(6497,\cdot)\)
\(\chi_{152269}(6829,\cdot)\)
\(\chi_{152269}(7349,\cdot)\)
\(\chi_{152269}(7895,\cdot)\)
\(\chi_{152269}(9786,\cdot)\)
\(\chi_{152269}(9897,\cdot)\)
\(\chi_{152269}(11580,\cdot)\)
\(\chi_{152269}(13348,\cdot)\)
\(\chi_{152269}(14011,\cdot)\)
\(\chi_{152269}(15123,\cdot)\)
\(\chi_{152269}(16306,\cdot)\)
\(\chi_{152269}(17177,\cdot)\)
\(\chi_{152269}(18197,\cdot)\)
\(\chi_{152269}(20784,\cdot)\)
\(\chi_{152269}(22175,\cdot)\)
\(\chi_{152269}(22910,\cdot)\)
\(\chi_{152269}(26160,\cdot)\)
\(\chi_{152269}(27700,\cdot)\)
...
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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 };
\((149567,143313,83318)\) → \((e\left(\frac{41}{78}\right),e\left(\frac{1}{16}\right),e\left(\frac{37}{52}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 152269 }(14011, a) \) |
\(1\) | \(1\) | \(e\left(\frac{35}{312}\right)\) | \(e\left(\frac{211}{624}\right)\) | \(e\left(\frac{35}{156}\right)\) | \(e\left(\frac{101}{208}\right)\) | \(e\left(\frac{281}{624}\right)\) | \(e\left(\frac{557}{624}\right)\) | \(e\left(\frac{35}{104}\right)\) | \(e\left(\frac{211}{312}\right)\) | \(e\left(\frac{373}{624}\right)\) | \(e\left(\frac{529}{624}\right)\) |
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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)
