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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(21775, base_ring=CyclotomicField(330))
M = H._module
chi = DirichletCharacter(H, M([231,55,265]))
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gp:[g,chi] = znchar(Mod(5334, 21775))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("21775.5334");
| Modulus: | \(21775\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(21775\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(330\) |
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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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\(\chi_{21775}(264,\cdot)\)
\(\chi_{21775}(979,\cdot)\)
\(\chi_{21775}(1219,\cdot)\)
\(\chi_{21775}(1304,\cdot)\)
\(\chi_{21775}(1954,\cdot)\)
\(\chi_{21775}(2084,\cdot)\)
\(\chi_{21775}(2194,\cdot)\)
\(\chi_{21775}(2259,\cdot)\)
\(\chi_{21775}(2389,\cdot)\)
\(\chi_{21775}(2909,\cdot)\)
\(\chi_{21775}(2994,\cdot)\)
\(\chi_{21775}(3234,\cdot)\)
\(\chi_{21775}(3384,\cdot)\)
\(\chi_{21775}(3754,\cdot)\)
\(\chi_{21775}(3839,\cdot)\)
\(\chi_{21775}(4014,\cdot)\)
\(\chi_{21775}(4144,\cdot)\)
\(\chi_{21775}(4339,\cdot)\)
\(\chi_{21775}(4619,\cdot)\)
\(\chi_{21775}(5334,\cdot)\)
\(\chi_{21775}(5659,\cdot)\)
\(\chi_{21775}(6309,\cdot)\)
\(\chi_{21775}(6439,\cdot)\)
\(\chi_{21775}(6614,\cdot)\)
\(\chi_{21775}(6744,\cdot)\)
\(\chi_{21775}(7264,\cdot)\)
\(\chi_{21775}(7589,\cdot)\)
\(\chi_{21775}(7739,\cdot)\)
\(\chi_{21775}(7804,\cdot)\)
\(\chi_{21775}(8109,\cdot)\)
...
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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 };
\((5227,10051,6501)\) → \((e\left(\frac{7}{10}\right),e\left(\frac{1}{6}\right),e\left(\frac{53}{66}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(14\) |
| \( \chi_{ 21775 }(5334, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{221}{330}\right)\) | \(e\left(\frac{146}{165}\right)\) | \(e\left(\frac{56}{165}\right)\) | \(e\left(\frac{61}{110}\right)\) | \(e\left(\frac{53}{66}\right)\) | \(e\left(\frac{1}{110}\right)\) | \(e\left(\frac{127}{165}\right)\) | \(e\left(\frac{41}{55}\right)\) | \(e\left(\frac{37}{165}\right)\) | \(e\left(\frac{26}{55}\right)\) |
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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)
