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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(109744, base_ring=CyclotomicField(1444))
M = H._module
chi = DirichletCharacter(H, M([0,1083,1138]))
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gp:[g,chi] = znchar(Mod(42445, 109744))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("109744.42445");
| Modulus: | \(109744\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(109744\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(1444\) |
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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);
|
\(\chi_{109744}(37,\cdot)\)
\(\chi_{109744}(189,\cdot)\)
\(\chi_{109744}(341,\cdot)\)
\(\chi_{109744}(493,\cdot)\)
\(\chi_{109744}(645,\cdot)\)
\(\chi_{109744}(797,\cdot)\)
\(\chi_{109744}(949,\cdot)\)
\(\chi_{109744}(1101,\cdot)\)
\(\chi_{109744}(1253,\cdot)\)
\(\chi_{109744}(1405,\cdot)\)
\(\chi_{109744}(1557,\cdot)\)
\(\chi_{109744}(1709,\cdot)\)
\(\chi_{109744}(1861,\cdot)\)
\(\chi_{109744}(2013,\cdot)\)
\(\chi_{109744}(2317,\cdot)\)
\(\chi_{109744}(2469,\cdot)\)
\(\chi_{109744}(2621,\cdot)\)
\(\chi_{109744}(2773,\cdot)\)
\(\chi_{109744}(2925,\cdot)\)
\(\chi_{109744}(3077,\cdot)\)
\(\chi_{109744}(3229,\cdot)\)
\(\chi_{109744}(3381,\cdot)\)
\(\chi_{109744}(3533,\cdot)\)
\(\chi_{109744}(3685,\cdot)\)
\(\chi_{109744}(3837,\cdot)\)
\(\chi_{109744}(3989,\cdot)\)
\(\chi_{109744}(4141,\cdot)\)
\(\chi_{109744}(4293,\cdot)\)
\(\chi_{109744}(4445,\cdot)\)
\(\chi_{109744}(4597,\cdot)\)
...
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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 };
\((68591,82309,89169)\) → \((1,-i,e\left(\frac{569}{722}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(21\) | \(23\) |
| \( \chi_{ 109744 }(42445, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{1147}{1444}\right)\) | \(e\left(\frac{1379}{1444}\right)\) | \(e\left(\frac{173}{722}\right)\) | \(e\left(\frac{425}{722}\right)\) | \(e\left(\frac{119}{1444}\right)\) | \(e\left(\frac{615}{1444}\right)\) | \(e\left(\frac{541}{722}\right)\) | \(e\left(\frac{210}{361}\right)\) | \(e\left(\frac{49}{1444}\right)\) | \(e\left(\frac{253}{722}\right)\) |
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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)
