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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4961, base_ring=CyclotomicField(110))
M = H._module
chi = DirichletCharacter(H, M([5,99]))
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gp:[g,chi] = znchar(Mod(3057, 4961))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("4961.3057");
| Modulus: | \(4961\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(4961\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(110\) |
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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);
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\(\chi_{4961}(230,\cdot)\)
\(\chi_{4961}(318,\cdot)\)
\(\chi_{4961}(351,\cdot)\)
\(\chi_{4961}(373,\cdot)\)
\(\chi_{4961}(681,\cdot)\)
\(\chi_{4961}(769,\cdot)\)
\(\chi_{4961}(802,\cdot)\)
\(\chi_{4961}(824,\cdot)\)
\(\chi_{4961}(1132,\cdot)\)
\(\chi_{4961}(1220,\cdot)\)
\(\chi_{4961}(1253,\cdot)\)
\(\chi_{4961}(1275,\cdot)\)
\(\chi_{4961}(1583,\cdot)\)
\(\chi_{4961}(1671,\cdot)\)
\(\chi_{4961}(1704,\cdot)\)
\(\chi_{4961}(1726,\cdot)\)
\(\chi_{4961}(2034,\cdot)\)
\(\chi_{4961}(2122,\cdot)\)
\(\chi_{4961}(2155,\cdot)\)
\(\chi_{4961}(2485,\cdot)\)
\(\chi_{4961}(2573,\cdot)\)
\(\chi_{4961}(2606,\cdot)\)
\(\chi_{4961}(2628,\cdot)\)
\(\chi_{4961}(2936,\cdot)\)
\(\chi_{4961}(3057,\cdot)\)
\(\chi_{4961}(3079,\cdot)\)
\(\chi_{4961}(3475,\cdot)\)
\(\chi_{4961}(3530,\cdot)\)
\(\chi_{4961}(3838,\cdot)\)
\(\chi_{4961}(3926,\cdot)\)
...
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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 };
\((2543,2179)\) → \((e\left(\frac{1}{22}\right),e\left(\frac{9}{10}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(12\) |
| \( \chi_{ 4961 }(3057, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{49}{110}\right)\) | \(-1\) | \(e\left(\frac{49}{55}\right)\) | \(e\left(\frac{9}{55}\right)\) | \(e\left(\frac{52}{55}\right)\) | \(e\left(\frac{23}{55}\right)\) | \(e\left(\frac{37}{110}\right)\) | \(1\) | \(e\left(\frac{67}{110}\right)\) | \(e\left(\frac{43}{110}\right)\) |
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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)
