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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4163, base_ring=CyclotomicField(990))
M = H._module
chi = DirichletCharacter(H, M([360,737]))
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gp:[g,chi] = znchar(Mod(1005, 4163))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("4163.1005");
| 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: | \(990\) |
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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);
|
\(\chi_{4163}(4,\cdot)\)
\(\chi_{4163}(12,\cdot)\)
\(\chi_{4163}(52,\cdot)\)
\(\chi_{4163}(55,\cdot)\)
\(\chi_{4163}(94,\cdot)\)
\(\chi_{4163}(100,\cdot)\)
\(\chi_{4163}(147,\cdot)\)
\(\chi_{4163}(165,\cdot)\)
\(\chi_{4163}(167,\cdot)\)
\(\chi_{4163}(192,\cdot)\)
\(\chi_{4163}(193,\cdot)\)
\(\chi_{4163}(233,\cdot)\)
\(\chi_{4163}(236,\cdot)\)
\(\chi_{4163}(292,\cdot)\)
\(\chi_{4163}(317,\cdot)\)
\(\chi_{4163}(324,\cdot)\)
\(\chi_{4163}(328,\cdot)\)
\(\chi_{4163}(347,\cdot)\)
\(\chi_{4163}(348,\cdot)\)
\(\chi_{4163}(349,\cdot)\)
\(\chi_{4163}(353,\cdot)\)
\(\chi_{4163}(374,\cdot)\)
\(\chi_{4163}(395,\cdot)\)
\(\chi_{4163}(399,\cdot)\)
\(\chi_{4163}(417,\cdot)\)
\(\chi_{4163}(422,\cdot)\)
\(\chi_{4163}(441,\cdot)\)
\(\chi_{4163}(462,\cdot)\)
\(\chi_{4163}(468,\cdot)\)
\(\chi_{4163}(473,\cdot)\)
...
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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 };
\((2535,1450)\) → \((e\left(\frac{4}{11}\right),e\left(\frac{67}{90}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 4163 }(1005, a) \) |
\(1\) | \(1\) | \(e\left(\frac{467}{990}\right)\) | \(e\left(\frac{251}{495}\right)\) | \(e\left(\frac{467}{495}\right)\) | \(e\left(\frac{82}{165}\right)\) | \(e\left(\frac{323}{330}\right)\) | \(e\left(\frac{5}{66}\right)\) | \(e\left(\frac{137}{330}\right)\) | \(e\left(\frac{7}{495}\right)\) | \(e\left(\frac{959}{990}\right)\) | \(e\left(\frac{212}{495}\right)\) |
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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)
