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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4699, base_ring=CyclotomicField(126))
M = H._module
chi = DirichletCharacter(H, M([56,43]))
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gp:[g,chi] = znchar(Mod(194, 4699))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("4699.194");
| Modulus: | \(4699\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(4699\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(126\) |
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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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\(\chi_{4699}(12,\cdot)\)
\(\chi_{4699}(194,\cdot)\)
\(\chi_{4699}(218,\cdot)\)
\(\chi_{4699}(268,\cdot)\)
\(\chi_{4699}(564,\cdot)\)
\(\chi_{4699}(604,\cdot)\)
\(\chi_{4699}(641,\cdot)\)
\(\chi_{4699}(848,\cdot)\)
\(\chi_{4699}(937,\cdot)\)
\(\chi_{4699}(1200,\cdot)\)
\(\chi_{4699}(1440,\cdot)\)
\(\chi_{4699}(1570,\cdot)\)
\(\chi_{4699}(1709,\cdot)\)
\(\chi_{4699}(1736,\cdot)\)
\(\chi_{4699}(1748,\cdot)\)
\(\chi_{4699}(1785,\cdot)\)
\(\chi_{4699}(1884,\cdot)\)
\(\chi_{4699}(1894,\cdot)\)
\(\chi_{4699}(1958,\cdot)\)
\(\chi_{4699}(2142,\cdot)\)
\(\chi_{4699}(2273,\cdot)\)
\(\chi_{4699}(2525,\cdot)\)
\(\chi_{4699}(2745,\cdot)\)
\(\chi_{4699}(3004,\cdot)\)
\(\chi_{4699}(3013,\cdot)\)
\(\chi_{4699}(3030,\cdot)\)
\(\chi_{4699}(3141,\cdot)\)
\(\chi_{4699}(3198,\cdot)\)
\(\chi_{4699}(3305,\cdot)\)
\(\chi_{4699}(3966,\cdot)\)
...
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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 };
\((890,2924)\) → \((e\left(\frac{4}{9}\right),e\left(\frac{43}{126}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 4699 }(194, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{1}{63}\right)\) | \(e\left(\frac{113}{126}\right)\) | \(e\left(\frac{2}{63}\right)\) | \(e\left(\frac{115}{126}\right)\) | \(e\left(\frac{115}{126}\right)\) | \(e\left(\frac{59}{126}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{50}{63}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{34}{63}\right)\) |
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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)
