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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(12221, base_ring=CyclotomicField(1100))
M = H._module
chi = DirichletCharacter(H, M([680,363]))
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gp:[g,chi] = znchar(Mod(641, 12221))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("12221.641");
| Modulus: | \(12221\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(12221\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(1100\) |
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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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\(\chi_{12221}(26,\cdot)\)
\(\chi_{12221}(104,\cdot)\)
\(\chi_{12221}(147,\cdot)\)
\(\chi_{12221}(174,\cdot)\)
\(\chi_{12221}(257,\cdot)\)
\(\chi_{12221}(311,\cdot)\)
\(\chi_{12221}(378,\cdot)\)
\(\chi_{12221}(401,\cdot)\)
\(\chi_{12221}(416,\cdot)\)
\(\chi_{12221}(438,\cdot)\)
\(\chi_{12221}(471,\cdot)\)
\(\chi_{12221}(520,\cdot)\)
\(\chi_{12221}(532,\cdot)\)
\(\chi_{12221}(543,\cdot)\)
\(\chi_{12221}(553,\cdot)\)
\(\chi_{12221}(555,\cdot)\)
\(\chi_{12221}(588,\cdot)\)
\(\chi_{12221}(599,\cdot)\)
\(\chi_{12221}(641,\cdot)\)
\(\chi_{12221}(665,\cdot)\)
\(\chi_{12221}(696,\cdot)\)
\(\chi_{12221}(718,\cdot)\)
\(\chi_{12221}(773,\cdot)\)
\(\chi_{12221}(779,\cdot)\)
\(\chi_{12221}(861,\cdot)\)
\(\chi_{12221}(894,\cdot)\)
\(\chi_{12221}(911,\cdot)\)
\(\chi_{12221}(951,\cdot)\)
\(\chi_{12221}(972,\cdot)\)
\(\chi_{12221}(983,\cdot)\)
...
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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 };
\((607,11617)\) → \((e\left(\frac{34}{55}\right),e\left(\frac{33}{100}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(12\) |
| \( \chi_{ 12221 }(641, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{1043}{1100}\right)\) | \(e\left(\frac{17}{100}\right)\) | \(e\left(\frac{493}{550}\right)\) | \(e\left(\frac{183}{275}\right)\) | \(e\left(\frac{13}{110}\right)\) | \(e\left(\frac{327}{1100}\right)\) | \(e\left(\frac{929}{1100}\right)\) | \(e\left(\frac{17}{50}\right)\) | \(e\left(\frac{27}{44}\right)\) | \(e\left(\frac{73}{1100}\right)\) |
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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)
