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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(59245, base_ring=CyclotomicField(136))
M = H._module
chi = DirichletCharacter(H, M([102,86,17]))
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gp:[g,chi] = znchar(Mod(21798, 59245))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("59245.21798");
| Modulus: | \(59245\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(59245\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(136\) |
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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);
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\(\chi_{59245}(888,\cdot)\)
\(\chi_{59245}(2843,\cdot)\)
\(\chi_{59245}(3277,\cdot)\)
\(\chi_{59245}(3447,\cdot)\)
\(\chi_{59245}(6328,\cdot)\)
\(\chi_{59245}(6762,\cdot)\)
\(\chi_{59245}(6932,\cdot)\)
\(\chi_{59245}(7858,\cdot)\)
\(\chi_{59245}(9813,\cdot)\)
\(\chi_{59245}(10247,\cdot)\)
\(\chi_{59245}(10417,\cdot)\)
\(\chi_{59245}(11343,\cdot)\)
\(\chi_{59245}(13298,\cdot)\)
\(\chi_{59245}(13732,\cdot)\)
\(\chi_{59245}(13902,\cdot)\)
\(\chi_{59245}(14828,\cdot)\)
\(\chi_{59245}(16783,\cdot)\)
\(\chi_{59245}(17217,\cdot)\)
\(\chi_{59245}(17387,\cdot)\)
\(\chi_{59245}(18313,\cdot)\)
\(\chi_{59245}(20702,\cdot)\)
\(\chi_{59245}(20872,\cdot)\)
\(\chi_{59245}(21798,\cdot)\)
\(\chi_{59245}(23753,\cdot)\)
\(\chi_{59245}(24187,\cdot)\)
\(\chi_{59245}(24357,\cdot)\)
\(\chi_{59245}(25283,\cdot)\)
\(\chi_{59245}(27238,\cdot)\)
\(\chi_{59245}(27672,\cdot)\)
\(\chi_{59245}(27842,\cdot)\)
...
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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 };
\((47397,35261,30346)\) → \((-i,e\left(\frac{43}{68}\right),e\left(\frac{1}{8}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
| \( \chi_{ 59245 }(21798, a) \) |
\(1\) | \(1\) | \(e\left(\frac{5}{34}\right)\) | \(e\left(\frac{103}{136}\right)\) | \(e\left(\frac{5}{17}\right)\) | \(e\left(\frac{123}{136}\right)\) | \(e\left(\frac{87}{136}\right)\) | \(e\left(\frac{15}{34}\right)\) | \(e\left(\frac{35}{68}\right)\) | \(e\left(\frac{125}{136}\right)\) | \(e\left(\frac{7}{136}\right)\) | \(e\left(\frac{9}{136}\right)\) |
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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)
