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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2404, base_ring=CyclotomicField(600))
M = H._module
chi = DirichletCharacter(H, M([300,331]))
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gp:[g,chi] = znchar(Mod(1295, 2404))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2404.1295");
| Modulus: | \(2404\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(2404\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(600\) |
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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);
|
\(\chi_{2404}(7,\cdot)\)
\(\chi_{2404}(11,\cdot)\)
\(\chi_{2404}(19,\cdot)\)
\(\chi_{2404}(35,\cdot)\)
\(\chi_{2404}(43,\cdot)\)
\(\chi_{2404}(55,\cdot)\)
\(\chi_{2404}(91,\cdot)\)
\(\chi_{2404}(95,\cdot)\)
\(\chi_{2404}(103,\cdot)\)
\(\chi_{2404}(107,\cdot)\)
\(\chi_{2404}(127,\cdot)\)
\(\chi_{2404}(143,\cdot)\)
\(\chi_{2404}(155,\cdot)\)
\(\chi_{2404}(159,\cdot)\)
\(\chi_{2404}(247,\cdot)\)
\(\chi_{2404}(255,\cdot)\)
\(\chi_{2404}(283,\cdot)\)
\(\chi_{2404}(291,\cdot)\)
\(\chi_{2404}(311,\cdot)\)
\(\chi_{2404}(315,\cdot)\)
\(\chi_{2404}(327,\cdot)\)
\(\chi_{2404}(347,\cdot)\)
\(\chi_{2404}(395,\cdot)\)
\(\chi_{2404}(411,\cdot)\)
\(\chi_{2404}(415,\cdot)\)
\(\chi_{2404}(419,\cdot)\)
\(\chi_{2404}(427,\cdot)\)
\(\chi_{2404}(455,\cdot)\)
\(\chi_{2404}(459,\cdot)\)
\(\chi_{2404}(487,\cdot)\)
...
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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 };
\((1203,1209)\) → \((-1,e\left(\frac{331}{600}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 2404 }(1295, a) \) |
\(1\) | \(1\) | \(e\left(\frac{31}{150}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{31}{600}\right)\) | \(e\left(\frac{31}{75}\right)\) | \(e\left(\frac{67}{600}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{187}{300}\right)\) | \(e\left(\frac{67}{120}\right)\) | \(e\left(\frac{271}{600}\right)\) | \(e\left(\frac{31}{120}\right)\) |
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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)
