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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1843, base_ring=CyclotomicField(144))
M = H._module
chi = DirichletCharacter(H, M([136,51]))
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gp:[g,chi] = znchar(Mod(390, 1843))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1843.390");
| Modulus: | \(1843\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(1843\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(144\) |
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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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\(\chi_{1843}(53,\cdot)\)
\(\chi_{1843}(238,\cdot)\)
\(\chi_{1843}(242,\cdot)\)
\(\chi_{1843}(260,\cdot)\)
\(\chi_{1843}(280,\cdot)\)
\(\chi_{1843}(356,\cdot)\)
\(\chi_{1843}(357,\cdot)\)
\(\chi_{1843}(363,\cdot)\)
\(\chi_{1843}(390,\cdot)\)
\(\chi_{1843}(413,\cdot)\)
\(\chi_{1843}(420,\cdot)\)
\(\chi_{1843}(488,\cdot)\)
\(\chi_{1843}(496,\cdot)\)
\(\chi_{1843}(534,\cdot)\)
\(\chi_{1843}(580,\cdot)\)
\(\chi_{1843}(585,\cdot)\)
\(\chi_{1843}(630,\cdot)\)
\(\chi_{1843}(668,\cdot)\)
\(\chi_{1843}(732,\cdot)\)
\(\chi_{1843}(744,\cdot)\)
\(\chi_{1843}(773,\cdot)\)
\(\chi_{1843}(801,\cdot)\)
\(\chi_{1843}(870,\cdot)\)
\(\chi_{1843}(945,\cdot)\)
\(\chi_{1843}(1001,\cdot)\)
\(\chi_{1843}(1002,\cdot)\)
\(\chi_{1843}(1036,\cdot)\)
\(\chi_{1843}(1078,\cdot)\)
\(\chi_{1843}(1098,\cdot)\)
\(\chi_{1843}(1116,\cdot)\)
...
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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 };
\((971,1654)\) → \((e\left(\frac{17}{18}\right),e\left(\frac{17}{48}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 1843 }(390, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{71}{72}\right)\) | \(e\left(\frac{5}{72}\right)\) | \(e\left(\frac{35}{36}\right)\) | \(e\left(\frac{67}{144}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{31}{48}\right)\) | \(e\left(\frac{23}{24}\right)\) | \(e\left(\frac{5}{36}\right)\) | \(e\left(\frac{65}{144}\right)\) | \(e\left(\frac{19}{24}\right)\) |
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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)
