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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(220669, base_ring=CyclotomicField(296))
M = H._module
chi = DirichletCharacter(H, M([104,101]))
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gp:[g,chi] = znchar(Mod(34500, 220669))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("220669.34500");
| 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: | \(296\) |
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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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\(\chi_{220669}(770,\cdot)\)
\(\chi_{220669}(2360,\cdot)\)
\(\chi_{220669}(3401,\cdot)\)
\(\chi_{220669}(4864,\cdot)\)
\(\chi_{220669}(7426,\cdot)\)
\(\chi_{220669}(8302,\cdot)\)
\(\chi_{220669}(11859,\cdot)\)
\(\chi_{220669}(12249,\cdot)\)
\(\chi_{220669}(12384,\cdot)\)
\(\chi_{220669}(13086,\cdot)\)
\(\chi_{220669}(15261,\cdot)\)
\(\chi_{220669}(15521,\cdot)\)
\(\chi_{220669}(19685,\cdot)\)
\(\chi_{220669}(20517,\cdot)\)
\(\chi_{220669}(20593,\cdot)\)
\(\chi_{220669}(23384,\cdot)\)
\(\chi_{220669}(23627,\cdot)\)
\(\chi_{220669}(24753,\cdot)\)
\(\chi_{220669}(25621,\cdot)\)
\(\chi_{220669}(27036,\cdot)\)
\(\chi_{220669}(27571,\cdot)\)
\(\chi_{220669}(29237,\cdot)\)
\(\chi_{220669}(29846,\cdot)\)
\(\chi_{220669}(33084,\cdot)\)
\(\chi_{220669}(33755,\cdot)\)
\(\chi_{220669}(34500,\cdot)\)
\(\chi_{220669}(35602,\cdot)\)
\(\chi_{220669}(52751,\cdot)\)
\(\chi_{220669}(53090,\cdot)\)
\(\chi_{220669}(56653,\cdot)\)
...
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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 };
\((48874,122926)\) → \((e\left(\frac{13}{37}\right),e\left(\frac{101}{296}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 220669 }(34500, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{1}{74}\right)\) | \(e\left(\frac{269}{296}\right)\) | \(e\left(\frac{1}{37}\right)\) | \(e\left(\frac{3}{148}\right)\) | \(e\left(\frac{273}{296}\right)\) | \(e\left(\frac{30}{37}\right)\) | \(e\left(\frac{3}{74}\right)\) | \(e\left(\frac{121}{148}\right)\) | \(e\left(\frac{5}{148}\right)\) | \(e\left(\frac{35}{296}\right)\) |
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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)
