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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4181, base_ring=CyclotomicField(1008))
M = H._module
chi = DirichletCharacter(H, M([644,495]))
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gp:[g,chi] = znchar(Mod(301, 4181))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("4181.301");
| Modulus: | \(4181\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(4181\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(1008\) |
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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);
|
\(\chi_{4181}(5,\cdot)\)
\(\chi_{4181}(19,\cdot)\)
\(\chi_{4181}(24,\cdot)\)
\(\chi_{4181}(54,\cdot)\)
\(\chi_{4181}(55,\cdot)\)
\(\chi_{4181}(59,\cdot)\)
\(\chi_{4181}(89,\cdot)\)
\(\chi_{4181}(92,\cdot)\)
\(\chi_{4181}(93,\cdot)\)
\(\chi_{4181}(133,\cdot)\)
\(\chi_{4181}(146,\cdot)\)
\(\chi_{4181}(167,\cdot)\)
\(\chi_{4181}(168,\cdot)\)
\(\chi_{4181}(180,\cdot)\)
\(\chi_{4181}(183,\cdot)\)
\(\chi_{4181}(203,\cdot)\)
\(\chi_{4181}(205,\cdot)\)
\(\chi_{4181}(207,\cdot)\)
\(\chi_{4181}(209,\cdot)\)
\(\chi_{4181}(264,\cdot)\)
\(\chi_{4181}(301,\cdot)\)
\(\chi_{4181}(316,\cdot)\)
\(\chi_{4181}(351,\cdot)\)
\(\chi_{4181}(385,\cdot)\)
\(\chi_{4181}(394,\cdot)\)
\(\chi_{4181}(405,\cdot)\)
\(\chi_{4181}(431,\cdot)\)
\(\chi_{4181}(442,\cdot)\)
\(\chi_{4181}(446,\cdot)\)
\(\chi_{4181}(457,\cdot)\)
...
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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 };
\((2148,4071)\) → \((e\left(\frac{23}{36}\right),e\left(\frac{55}{112}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 4181 }(301, a) \) |
\(1\) | \(1\) | \(e\left(\frac{67}{126}\right)\) | \(e\left(\frac{103}{1008}\right)\) | \(e\left(\frac{4}{63}\right)\) | \(e\left(\frac{457}{1008}\right)\) | \(e\left(\frac{71}{112}\right)\) | \(e\left(\frac{47}{126}\right)\) | \(e\left(\frac{25}{42}\right)\) | \(e\left(\frac{103}{504}\right)\) | \(e\left(\frac{331}{336}\right)\) | \(e\left(\frac{85}{168}\right)\) |
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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)
