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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(91723, base_ring=CyclotomicField(666))
M = H._module
chi = DirichletCharacter(H, M([605,222]))
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gp:[g,chi] = znchar(Mod(4794, 91723))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("91723.4794");
| Modulus: | \(91723\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(91723\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(666\) |
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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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\(\chi_{91723}(506,\cdot)\)
\(\chi_{91723}(1436,\cdot)\)
\(\chi_{91723}(2039,\cdot)\)
\(\chi_{91723}(2248,\cdot)\)
\(\chi_{91723}(2315,\cdot)\)
\(\chi_{91723}(2985,\cdot)\)
\(\chi_{91723}(3111,\cdot)\)
\(\chi_{91723}(3915,\cdot)\)
\(\chi_{91723}(4518,\cdot)\)
\(\chi_{91723}(4727,\cdot)\)
\(\chi_{91723}(4794,\cdot)\)
\(\chi_{91723}(5464,\cdot)\)
\(\chi_{91723}(5590,\cdot)\)
\(\chi_{91723}(6394,\cdot)\)
\(\chi_{91723}(6997,\cdot)\)
\(\chi_{91723}(7206,\cdot)\)
\(\chi_{91723}(7273,\cdot)\)
\(\chi_{91723}(7943,\cdot)\)
\(\chi_{91723}(8069,\cdot)\)
\(\chi_{91723}(8873,\cdot)\)
\(\chi_{91723}(9476,\cdot)\)
\(\chi_{91723}(9685,\cdot)\)
\(\chi_{91723}(9752,\cdot)\)
\(\chi_{91723}(10422,\cdot)\)
\(\chi_{91723}(10548,\cdot)\)
\(\chi_{91723}(11352,\cdot)\)
\(\chi_{91723}(11955,\cdot)\)
\(\chi_{91723}(12164,\cdot)\)
\(\chi_{91723}(12231,\cdot)\)
\(\chi_{91723}(12901,\cdot)\)
...
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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 };
\((41072,50654)\) → \((e\left(\frac{605}{666}\right),e\left(\frac{1}{3}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 91723 }(4794, a) \) |
\(1\) | \(1\) | \(e\left(\frac{161}{666}\right)\) | \(e\left(\frac{80}{333}\right)\) | \(e\left(\frac{161}{333}\right)\) | \(e\left(\frac{55}{666}\right)\) | \(e\left(\frac{107}{222}\right)\) | \(e\left(\frac{47}{333}\right)\) | \(e\left(\frac{161}{222}\right)\) | \(e\left(\frac{160}{333}\right)\) | \(e\left(\frac{12}{37}\right)\) | \(e\left(\frac{21}{37}\right)\) |
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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)
