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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9499, base_ring=CyclotomicField(174))
M = H._module
chi = DirichletCharacter(H, M([29,87,24]))
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gp:[g,chi] = znchar(Mod(3265, 9499))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("9499.3265");
| Modulus: | \(9499\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(9499\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(174\) |
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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);
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\(\chi_{9499}(45,\cdot)\)
\(\chi_{9499}(68,\cdot)\)
\(\chi_{9499}(206,\cdot)\)
\(\chi_{9499}(390,\cdot)\)
\(\chi_{9499}(551,\cdot)\)
\(\chi_{9499}(712,\cdot)\)
\(\chi_{9499}(1172,\cdot)\)
\(\chi_{9499}(1195,\cdot)\)
\(\chi_{9499}(1333,\cdot)\)
\(\chi_{9499}(1494,\cdot)\)
\(\chi_{9499}(1655,\cdot)\)
\(\chi_{9499}(1678,\cdot)\)
\(\chi_{9499}(1816,\cdot)\)
\(\chi_{9499}(2000,\cdot)\)
\(\chi_{9499}(2299,\cdot)\)
\(\chi_{9499}(2322,\cdot)\)
\(\chi_{9499}(2460,\cdot)\)
\(\chi_{9499}(2483,\cdot)\)
\(\chi_{9499}(2621,\cdot)\)
\(\chi_{9499}(2644,\cdot)\)
\(\chi_{9499}(2782,\cdot)\)
\(\chi_{9499}(2966,\cdot)\)
\(\chi_{9499}(3104,\cdot)\)
\(\chi_{9499}(3265,\cdot)\)
\(\chi_{9499}(3426,\cdot)\)
\(\chi_{9499}(3449,\cdot)\)
\(\chi_{9499}(3909,\cdot)\)
\(\chi_{9499}(4093,\cdot)\)
\(\chi_{9499}(4392,\cdot)\)
\(\chi_{9499}(4415,\cdot)\)
...
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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 };
\((6786,7848,1772)\) → \((e\left(\frac{1}{6}\right),-1,e\left(\frac{4}{29}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
| \( \chi_{ 9499 }(3265, a) \) |
\(1\) | \(1\) | \(e\left(\frac{41}{87}\right)\) | \(e\left(\frac{11}{174}\right)\) | \(e\left(\frac{82}{87}\right)\) | \(e\left(\frac{14}{87}\right)\) | \(e\left(\frac{31}{58}\right)\) | \(e\left(\frac{12}{29}\right)\) | \(e\left(\frac{11}{87}\right)\) | \(e\left(\frac{55}{87}\right)\) | \(e\left(\frac{107}{174}\right)\) | \(e\left(\frac{1}{174}\right)\) |
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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)
