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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(19747, base_ring=CyclotomicField(420))
M = H._module
chi = DirichletCharacter(H, M([130,35,126]))
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gp:[g,chi] = znchar(Mod(990, 19747))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("19747.990");
| Modulus: | \(19747\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(19747\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(420\) |
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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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\(\chi_{19747}(201,\cdot)\)
\(\chi_{19747}(306,\cdot)\)
\(\chi_{19747}(488,\cdot)\)
\(\chi_{19747}(990,\cdot)\)
\(\chi_{19747}(1081,\cdot)\)
\(\chi_{19747}(1263,\cdot)\)
\(\chi_{19747}(1298,\cdot)\)
\(\chi_{19747}(1480,\cdot)\)
\(\chi_{19747}(1627,\cdot)\)
\(\chi_{19747}(2385,\cdot)\)
\(\chi_{19747}(2476,\cdot)\)
\(\chi_{19747}(2658,\cdot)\)
\(\chi_{19747}(3022,\cdot)\)
\(\chi_{19747}(3036,\cdot)\)
\(\chi_{19747}(3127,\cdot)\)
\(\chi_{19747}(3309,\cdot)\)
\(\chi_{19747}(3673,\cdot)\)
\(\chi_{19747}(3811,\cdot)\)
\(\chi_{19747}(4028,\cdot)\)
\(\chi_{19747}(4084,\cdot)\)
\(\chi_{19747}(4119,\cdot)\)
\(\chi_{19747}(4301,\cdot)\)
\(\chi_{19747}(4448,\cdot)\)
\(\chi_{19747}(4665,\cdot)\)
\(\chi_{19747}(5206,\cdot)\)
\(\chi_{19747}(5297,\cdot)\)
\(\chi_{19747}(5479,\cdot)\)
\(\chi_{19747}(5843,\cdot)\)
\(\chi_{19747}(5857,\cdot)\)
\(\chi_{19747}(6130,\cdot)\)
...
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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 };
\((7255,9115,14015)\) → \((e\left(\frac{13}{42}\right),e\left(\frac{1}{12}\right),e\left(\frac{3}{10}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
| \( \chi_{ 19747 }(990, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{139}{420}\right)\) | \(e\left(\frac{33}{35}\right)\) | \(e\left(\frac{139}{210}\right)\) | \(e\left(\frac{61}{84}\right)\) | \(e\left(\frac{23}{84}\right)\) | \(e\left(\frac{139}{140}\right)\) | \(e\left(\frac{31}{35}\right)\) | \(e\left(\frac{2}{35}\right)\) | \(e\left(\frac{121}{140}\right)\) | \(e\left(\frac{127}{210}\right)\) |
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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)
