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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([690,549]))
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gp:[g,chi] = znchar(Mod(3264, 220669))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("220669.3264");
| 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: | 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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\(\chi_{220669}(240,\cdot)\)
\(\chi_{220669}(301,\cdot)\)
\(\chi_{220669}(983,\cdot)\)
\(\chi_{220669}(2165,\cdot)\)
\(\chi_{220669}(2232,\cdot)\)
\(\chi_{220669}(2791,\cdot)\)
\(\chi_{220669}(2828,\cdot)\)
\(\chi_{220669}(2846,\cdot)\)
\(\chi_{220669}(3264,\cdot)\)
\(\chi_{220669}(3442,\cdot)\)
\(\chi_{220669}(4977,\cdot)\)
\(\chi_{220669}(5007,\cdot)\)
\(\chi_{220669}(5149,\cdot)\)
\(\chi_{220669}(5158,\cdot)\)
\(\chi_{220669}(5164,\cdot)\)
\(\chi_{220669}(5324,\cdot)\)
\(\chi_{220669}(5515,\cdot)\)
\(\chi_{220669}(5702,\cdot)\)
\(\chi_{220669}(6486,\cdot)\)
\(\chi_{220669}(7016,\cdot)\)
\(\chi_{220669}(7150,\cdot)\)
\(\chi_{220669}(7251,\cdot)\)
\(\chi_{220669}(8674,\cdot)\)
\(\chi_{220669}(9154,\cdot)\)
\(\chi_{220669}(9289,\cdot)\)
\(\chi_{220669}(9466,\cdot)\)
\(\chi_{220669}(9940,\cdot)\)
\(\chi_{220669}(10114,\cdot)\)
\(\chi_{220669}(10247,\cdot)\)
\(\chi_{220669}(10283,\cdot)\)
...
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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 };
\((48874,122926)\) → \((e\left(\frac{69}{148}\right),e\left(\frac{549}{1480}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 220669 }(3264, a) \) |
\(1\) | \(1\) | \(e\left(\frac{611}{740}\right)\) | \(e\left(\frac{1379}{1480}\right)\) | \(e\left(\frac{241}{370}\right)\) | \(e\left(\frac{151}{740}\right)\) | \(e\left(\frac{1121}{1480}\right)\) | \(e\left(\frac{249}{370}\right)\) | \(e\left(\frac{353}{740}\right)\) | \(e\left(\frac{639}{740}\right)\) | \(e\left(\frac{11}{370}\right)\) | \(e\left(\frac{209}{296}\right)\) |
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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)
