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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(29161, base_ring=CyclotomicField(660))
M = H._module
chi = DirichletCharacter(H, M([42,451]))
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gp:[g,chi] = znchar(Mod(4000, 29161))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("29161.4000");
| Modulus: | \(29161\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(29161\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(660\) |
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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);
|
\(\chi_{29161}(83,\cdot)\)
\(\chi_{29161}(90,\cdot)\)
\(\chi_{29161}(107,\cdot)\)
\(\chi_{29161}(145,\cdot)\)
\(\chi_{29161}(491,\cdot)\)
\(\chi_{29161}(579,\cdot)\)
\(\chi_{29161}(600,\cdot)\)
\(\chi_{29161}(1339,\cdot)\)
\(\chi_{29161}(1349,\cdot)\)
\(\chi_{29161}(1437,\cdot)\)
\(\chi_{29161}(1542,\cdot)\)
\(\chi_{29161}(1597,\cdot)\)
\(\chi_{29161}(1810,\cdot)\)
\(\chi_{29161}(2010,\cdot)\)
\(\chi_{29161}(2087,\cdot)\)
\(\chi_{29161}(2327,\cdot)\)
\(\chi_{29161}(2734,\cdot)\)
\(\chi_{29161}(2741,\cdot)\)
\(\chi_{29161}(2758,\cdot)\)
\(\chi_{29161}(2796,\cdot)\)
\(\chi_{29161}(3142,\cdot)\)
\(\chi_{29161}(3230,\cdot)\)
\(\chi_{29161}(3251,\cdot)\)
\(\chi_{29161}(4000,\cdot)\)
\(\chi_{29161}(4088,\cdot)\)
\(\chi_{29161}(4193,\cdot)\)
\(\chi_{29161}(4248,\cdot)\)
\(\chi_{29161}(4461,\cdot)\)
\(\chi_{29161}(4661,\cdot)\)
\(\chi_{29161}(4738,\cdot)\)
...
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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 };
\((28921,1453)\) → \((e\left(\frac{7}{110}\right),e\left(\frac{41}{60}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(12\) |
| \( \chi_{ 29161 }(4000, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{148}{165}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{131}{165}\right)\) | \(e\left(\frac{1}{110}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{17}{132}\right)\) | \(e\left(\frac{38}{55}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{299}{330}\right)\) | \(e\left(\frac{251}{330}\right)\) |
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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)
