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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(31753, base_ring=CyclotomicField(560))
M = H._module
chi = DirichletCharacter(H, M([15,188]))
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gp:[g,chi] = znchar(Mod(5564, 31753))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("31753.5564");
| Modulus: | \(31753\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(31753\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(560\) |
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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);
|
\(\chi_{31753}(209,\cdot)\)
\(\chi_{31753}(705,\cdot)\)
\(\chi_{31753}(825,\cdot)\)
\(\chi_{31753}(1798,\cdot)\)
\(\chi_{31753}(1818,\cdot)\)
\(\chi_{31753}(1950,\cdot)\)
\(\chi_{31753}(2127,\cdot)\)
\(\chi_{31753}(2176,\cdot)\)
\(\chi_{31753}(2231,\cdot)\)
\(\chi_{31753}(2328,\cdot)\)
\(\chi_{31753}(2354,\cdot)\)
\(\chi_{31753}(2457,\cdot)\)
\(\chi_{31753}(3034,\cdot)\)
\(\chi_{31753}(3352,\cdot)\)
\(\chi_{31753}(3622,\cdot)\)
\(\chi_{31753}(4198,\cdot)\)
\(\chi_{31753}(4424,\cdot)\)
\(\chi_{31753}(4553,\cdot)\)
\(\chi_{31753}(4610,\cdot)\)
\(\chi_{31753}(4645,\cdot)\)
\(\chi_{31753}(4791,\cdot)\)
\(\chi_{31753}(4902,\cdot)\)
\(\chi_{31753}(4926,\cdot)\)
\(\chi_{31753}(5001,\cdot)\)
\(\chi_{31753}(5218,\cdot)\)
\(\chi_{31753}(5282,\cdot)\)
\(\chi_{31753}(5564,\cdot)\)
\(\chi_{31753}(5611,\cdot)\)
\(\chi_{31753}(5629,\cdot)\)
\(\chi_{31753}(5689,\cdot)\)
...
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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 };
\((20795,10962)\) → \((e\left(\frac{3}{112}\right),e\left(\frac{47}{140}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 31753 }(5564, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{113}{140}\right)\) | \(e\left(\frac{29}{80}\right)\) | \(e\left(\frac{43}{70}\right)\) | \(e\left(\frac{373}{560}\right)\) | \(e\left(\frac{19}{112}\right)\) | \(e\left(\frac{11}{35}\right)\) | \(e\left(\frac{59}{140}\right)\) | \(e\left(\frac{29}{40}\right)\) | \(e\left(\frac{53}{112}\right)\) | \(e\left(\frac{257}{280}\right)\) |
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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)
