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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4163, base_ring=CyclotomicField(660))
M = H._module
chi = DirichletCharacter(H, M([630,583]))
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gp:[g,chi] = znchar(Mod(221, 4163))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("4163.221");
| Modulus: | \(4163\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(4163\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(660\) |
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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);
|
\(\chi_{4163}(30,\cdot)\)
\(\chi_{4163}(40,\cdot)\)
\(\chi_{4163}(51,\cdot)\)
\(\chi_{4163}(86,\cdot)\)
\(\chi_{4163}(113,\cdot)\)
\(\chi_{4163}(130,\cdot)\)
\(\chi_{4163}(175,\cdot)\)
\(\chi_{4163}(189,\cdot)\)
\(\chi_{4163}(221,\cdot)\)
\(\chi_{4163}(249,\cdot)\)
\(\chi_{4163}(267,\cdot)\)
\(\chi_{4163}(291,\cdot)\)
\(\chi_{4163}(332,\cdot)\)
\(\chi_{4163}(356,\cdot)\)
\(\chi_{4163}(402,\cdot)\)
\(\chi_{4163}(433,\cdot)\)
\(\chi_{4163}(448,\cdot)\)
\(\chi_{4163}(457,\cdot)\)
\(\chi_{4163}(475,\cdot)\)
\(\chi_{4163}(503,\cdot)\)
\(\chi_{4163}(513,\cdot)\)
\(\chi_{4163}(549,\cdot)\)
\(\chi_{4163}(573,\cdot)\)
\(\chi_{4163}(594,\cdot)\)
\(\chi_{4163}(638,\cdot)\)
\(\chi_{4163}(684,\cdot)\)
\(\chi_{4163}(718,\cdot)\)
\(\chi_{4163}(730,\cdot)\)
\(\chi_{4163}(732,\cdot)\)
\(\chi_{4163}(764,\cdot)\)
...
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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 };
\((2535,1450)\) → \((e\left(\frac{21}{22}\right),e\left(\frac{53}{60}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 4163 }(221, a) \) |
\(1\) | \(1\) | \(e\left(\frac{523}{660}\right)\) | \(e\left(\frac{122}{165}\right)\) | \(e\left(\frac{193}{330}\right)\) | \(e\left(\frac{83}{110}\right)\) | \(e\left(\frac{117}{220}\right)\) | \(e\left(\frac{17}{44}\right)\) | \(e\left(\frac{83}{220}\right)\) | \(e\left(\frac{79}{165}\right)\) | \(e\left(\frac{361}{660}\right)\) | \(e\left(\frac{59}{165}\right)\) |
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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)
