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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3143, base_ring=CyclotomicField(448))
M = H._module
chi = DirichletCharacter(H, M([224,429]))
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gp:[g,chi] = znchar(Mod(104, 3143))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("3143.104");
| Modulus: | \(3143\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(3143\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(448\) |
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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);
|
\(\chi_{3143}(6,\cdot)\)
\(\chi_{3143}(13,\cdot)\)
\(\chi_{3143}(27,\cdot)\)
\(\chi_{3143}(34,\cdot)\)
\(\chi_{3143}(48,\cdot)\)
\(\chi_{3143}(62,\cdot)\)
\(\chi_{3143}(69,\cdot)\)
\(\chi_{3143}(76,\cdot)\)
\(\chi_{3143}(83,\cdot)\)
\(\chi_{3143}(104,\cdot)\)
\(\chi_{3143}(132,\cdot)\)
\(\chi_{3143}(139,\cdot)\)
\(\chi_{3143}(146,\cdot)\)
\(\chi_{3143}(153,\cdot)\)
\(\chi_{3143}(216,\cdot)\)
\(\chi_{3143}(223,\cdot)\)
\(\chi_{3143}(272,\cdot)\)
\(\chi_{3143}(279,\cdot)\)
\(\chi_{3143}(286,\cdot)\)
\(\chi_{3143}(300,\cdot)\)
\(\chi_{3143}(307,\cdot)\)
\(\chi_{3143}(314,\cdot)\)
\(\chi_{3143}(342,\cdot)\)
\(\chi_{3143}(363,\cdot)\)
\(\chi_{3143}(384,\cdot)\)
\(\chi_{3143}(419,\cdot)\)
\(\chi_{3143}(461,\cdot)\)
\(\chi_{3143}(468,\cdot)\)
\(\chi_{3143}(475,\cdot)\)
\(\chi_{3143}(482,\cdot)\)
...
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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 };
\((899,2248)\) → \((-1,e\left(\frac{429}{448}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
| \( \chi_{ 3143 }(104, a) \) |
\(1\) | \(1\) | \(e\left(\frac{51}{224}\right)\) | \(e\left(\frac{205}{448}\right)\) | \(e\left(\frac{51}{112}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{307}{448}\right)\) | \(e\left(\frac{153}{224}\right)\) | \(e\left(\frac{205}{224}\right)\) | \(e\left(\frac{21}{32}\right)\) | \(e\left(\frac{45}{56}\right)\) | \(e\left(\frac{409}{448}\right)\) |
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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)
