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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(220669, base_ring=CyclotomicField(1480))
M = H._module
chi = DirichletCharacter(H, M([660,603]))
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gp:[g,chi] = znchar(Mod(7709, 220669))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("220669.7709");
| Modulus: | \(220669\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(220669\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(1480\) |
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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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\(\chi_{220669}(641,\cdot)\)
\(\chi_{220669}(814,\cdot)\)
\(\chi_{220669}(1234,\cdot)\)
\(\chi_{220669}(1292,\cdot)\)
\(\chi_{220669}(1350,\cdot)\)
\(\chi_{220669}(1376,\cdot)\)
\(\chi_{220669}(1684,\cdot)\)
\(\chi_{220669}(2559,\cdot)\)
\(\chi_{220669}(2578,\cdot)\)
\(\chi_{220669}(2649,\cdot)\)
\(\chi_{220669}(2795,\cdot)\)
\(\chi_{220669}(3211,\cdot)\)
\(\chi_{220669}(3402,\cdot)\)
\(\chi_{220669}(3751,\cdot)\)
\(\chi_{220669}(4241,\cdot)\)
\(\chi_{220669}(4986,\cdot)\)
\(\chi_{220669}(5347,\cdot)\)
\(\chi_{220669}(5806,\cdot)\)
\(\chi_{220669}(6118,\cdot)\)
\(\chi_{220669}(6191,\cdot)\)
\(\chi_{220669}(6972,\cdot)\)
\(\chi_{220669}(7295,\cdot)\)
\(\chi_{220669}(7411,\cdot)\)
\(\chi_{220669}(7709,\cdot)\)
\(\chi_{220669}(7872,\cdot)\)
\(\chi_{220669}(8107,\cdot)\)
\(\chi_{220669}(8763,\cdot)\)
\(\chi_{220669}(9053,\cdot)\)
\(\chi_{220669}(9636,\cdot)\)
\(\chi_{220669}(10603,\cdot)\)
...
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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 };
\((48874,122926)\) → \((e\left(\frac{33}{74}\right),e\left(\frac{603}{1480}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 220669 }(7709, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{13}{185}\right)\) | \(e\left(\frac{303}{1480}\right)\) | \(e\left(\frac{26}{185}\right)\) | \(e\left(\frac{657}{740}\right)\) | \(e\left(\frac{11}{40}\right)\) | \(e\left(\frac{104}{185}\right)\) | \(e\left(\frac{39}{185}\right)\) | \(e\left(\frac{303}{740}\right)\) | \(e\left(\frac{709}{740}\right)\) | \(e\left(\frac{1}{8}\right)\) |
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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)
