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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(29744, base_ring=CyclotomicField(260))
M = H._module
chi = DirichletCharacter(H, M([0,195,52,160]))
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gp:[g,chi] = znchar(Mod(9101, 29744))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("29744.9101");
| Modulus: | \(29744\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(29744\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(260\) |
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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);
|
\(\chi_{29744}(53,\cdot)\)
\(\chi_{29744}(157,\cdot)\)
\(\chi_{29744}(885,\cdot)\)
\(\chi_{29744}(1093,\cdot)\)
\(\chi_{29744}(1197,\cdot)\)
\(\chi_{29744}(1301,\cdot)\)
\(\chi_{29744}(2237,\cdot)\)
\(\chi_{29744}(2341,\cdot)\)
\(\chi_{29744}(2445,\cdot)\)
\(\chi_{29744}(3173,\cdot)\)
\(\chi_{29744}(3485,\cdot)\)
\(\chi_{29744}(3589,\cdot)\)
\(\chi_{29744}(4317,\cdot)\)
\(\chi_{29744}(4525,\cdot)\)
\(\chi_{29744}(4629,\cdot)\)
\(\chi_{29744}(5461,\cdot)\)
\(\chi_{29744}(5669,\cdot)\)
\(\chi_{29744}(5773,\cdot)\)
\(\chi_{29744}(5877,\cdot)\)
\(\chi_{29744}(6605,\cdot)\)
\(\chi_{29744}(6813,\cdot)\)
\(\chi_{29744}(6917,\cdot)\)
\(\chi_{29744}(7021,\cdot)\)
\(\chi_{29744}(7749,\cdot)\)
\(\chi_{29744}(7957,\cdot)\)
\(\chi_{29744}(8061,\cdot)\)
\(\chi_{29744}(8165,\cdot)\)
\(\chi_{29744}(8893,\cdot)\)
\(\chi_{29744}(9101,\cdot)\)
\(\chi_{29744}(9205,\cdot)\)
...
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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 };
\((18591,22309,13521,25521)\) → \((1,-i,e\left(\frac{1}{5}\right),e\left(\frac{8}{13}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(15\) | \(17\) | \(19\) | \(21\) | \(23\) | \(25\) |
| \( \chi_{ 29744 }(9101, a) \) |
\(1\) | \(1\) | \(e\left(\frac{41}{260}\right)\) | \(e\left(\frac{23}{260}\right)\) | \(e\left(\frac{97}{130}\right)\) | \(e\left(\frac{41}{130}\right)\) | \(e\left(\frac{16}{65}\right)\) | \(e\left(\frac{42}{65}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{47}{52}\right)\) | \(-1\) | \(e\left(\frac{23}{130}\right)\) |
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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)
