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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4307, base_ring=CyclotomicField(2088))
M = H._module
chi = DirichletCharacter(H, M([1728,319]))
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gp:[g,chi] = znchar(Mod(104, 4307))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("4307.104");
| Modulus: | \(4307\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(4307\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(2088\) |
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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);
|
\(\chi_{4307}(5,\cdot)\)
\(\chi_{4307}(15,\cdot)\)
\(\chi_{4307}(20,\cdot)\)
\(\chi_{4307}(26,\cdot)\)
\(\chi_{4307}(28,\cdot)\)
\(\chi_{4307}(29,\cdot)\)
\(\chi_{4307}(45,\cdot)\)
\(\chi_{4307}(53,\cdot)\)
\(\chi_{4307}(62,\cdot)\)
\(\chi_{4307}(68,\cdot)\)
\(\chi_{4307}(78,\cdot)\)
\(\chi_{4307}(84,\cdot)\)
\(\chi_{4307}(86,\cdot)\)
\(\chi_{4307}(87,\cdot)\)
\(\chi_{4307}(88,\cdot)\)
\(\chi_{4307}(104,\cdot)\)
\(\chi_{4307}(107,\cdot)\)
\(\chi_{4307}(112,\cdot)\)
\(\chi_{4307}(133,\cdot)\)
\(\chi_{4307}(135,\cdot)\)
\(\chi_{4307}(159,\cdot)\)
\(\chi_{4307}(166,\cdot)\)
\(\chi_{4307}(175,\cdot)\)
\(\chi_{4307}(180,\cdot)\)
\(\chi_{4307}(186,\cdot)\)
\(\chi_{4307}(193,\cdot)\)
\(\chi_{4307}(199,\cdot)\)
\(\chi_{4307}(204,\cdot)\)
\(\chi_{4307}(205,\cdot)\)
\(\chi_{4307}(206,\cdot)\)
...
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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 };
\((2775,1830)\) → \((e\left(\frac{24}{29}\right),e\left(\frac{11}{72}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 4307 }(104, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{13}{261}\right)\) | \(e\left(\frac{103}{348}\right)\) | \(e\left(\frac{26}{261}\right)\) | \(e\left(\frac{247}{2088}\right)\) | \(e\left(\frac{361}{1044}\right)\) | \(e\left(\frac{653}{696}\right)\) | \(e\left(\frac{13}{87}\right)\) | \(e\left(\frac{103}{174}\right)\) | \(e\left(\frac{39}{232}\right)\) | \(e\left(\frac{193}{2088}\right)\) |
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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)
