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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(224825, base_ring=CyclotomicField(20240))
M = H._module
chi = DirichletCharacter(H, M([11132,11385,19040]))
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gp:[g,chi] = znchar(Mod(473, 224825))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("224825.473");
| Modulus: | \(224825\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(224825\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(20240\) |
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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);
|
\(\chi_{224825}(3,\cdot)\)
\(\chi_{224825}(27,\cdot)\)
\(\chi_{224825}(48,\cdot)\)
\(\chi_{224825}(62,\cdot)\)
\(\chi_{224825}(73,\cdot)\)
\(\chi_{224825}(133,\cdot)\)
\(\chi_{224825}(142,\cdot)\)
\(\chi_{224825}(147,\cdot)\)
\(\chi_{224825}(173,\cdot)\)
\(\chi_{224825}(197,\cdot)\)
\(\chi_{224825}(233,\cdot)\)
\(\chi_{224825}(262,\cdot)\)
\(\chi_{224825}(303,\cdot)\)
\(\chi_{224825}(312,\cdot)\)
\(\chi_{224825}(317,\cdot)\)
\(\chi_{224825}(328,\cdot)\)
\(\chi_{224825}(347,\cdot)\)
\(\chi_{224825}(397,\cdot)\)
\(\chi_{224825}(403,\cdot)\)
\(\chi_{224825}(473,\cdot)\)
\(\chi_{224825}(537,\cdot)\)
\(\chi_{224825}(558,\cdot)\)
\(\chi_{224825}(583,\cdot)\)
\(\chi_{224825}(602,\cdot)\)
\(\chi_{224825}(652,\cdot)\)
\(\chi_{224825}(683,\cdot)\)
\(\chi_{224825}(742,\cdot)\)
\(\chi_{224825}(772,\cdot)\)
\(\chi_{224825}(813,\cdot)\)
\(\chi_{224825}(853,\cdot)\)
...
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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 };
\((62952,198376,25926)\) → \((e\left(\frac{11}{20}\right),e\left(\frac{9}{16}\right),e\left(\frac{238}{253}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
| \( \chi_{ 224825 }(473, a) \) |
\(1\) | \(1\) | \(e\left(\frac{5741}{10120}\right)\) | \(e\left(\frac{9389}{20240}\right)\) | \(e\left(\frac{681}{5060}\right)\) | \(e\left(\frac{631}{20240}\right)\) | \(e\left(\frac{1171}{4048}\right)\) | \(e\left(\frac{7103}{10120}\right)\) | \(e\left(\frac{9389}{10120}\right)\) | \(e\left(\frac{2367}{20240}\right)\) | \(e\left(\frac{12113}{20240}\right)\) | \(e\left(\frac{1101}{2530}\right)\) |
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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)
