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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5299, base_ring=CyclotomicField(756))
M = H._module
chi = DirichletCharacter(H, M([378,43]))
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gp:[g,chi] = znchar(Mod(1028, 5299))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("5299.1028");
| Modulus: | \(5299\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(5299\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(756\) |
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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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\(\chi_{5299}(6,\cdot)\)
\(\chi_{5299}(41,\cdot)\)
\(\chi_{5299}(97,\cdot)\)
\(\chi_{5299}(104,\cdot)\)
\(\chi_{5299}(111,\cdot)\)
\(\chi_{5299}(118,\cdot)\)
\(\chi_{5299}(139,\cdot)\)
\(\chi_{5299}(146,\cdot)\)
\(\chi_{5299}(188,\cdot)\)
\(\chi_{5299}(223,\cdot)\)
\(\chi_{5299}(279,\cdot)\)
\(\chi_{5299}(286,\cdot)\)
\(\chi_{5299}(293,\cdot)\)
\(\chi_{5299}(321,\cdot)\)
\(\chi_{5299}(370,\cdot)\)
\(\chi_{5299}(377,\cdot)\)
\(\chi_{5299}(391,\cdot)\)
\(\chi_{5299}(398,\cdot)\)
\(\chi_{5299}(405,\cdot)\)
\(\chi_{5299}(419,\cdot)\)
\(\chi_{5299}(447,\cdot)\)
\(\chi_{5299}(454,\cdot)\)
\(\chi_{5299}(503,\cdot)\)
\(\chi_{5299}(517,\cdot)\)
\(\chi_{5299}(608,\cdot)\)
\(\chi_{5299}(622,\cdot)\)
\(\chi_{5299}(650,\cdot)\)
\(\chi_{5299}(664,\cdot)\)
\(\chi_{5299}(720,\cdot)\)
\(\chi_{5299}(755,\cdot)\)
...
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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 };
\((1515,4544)\) → \((-1,e\left(\frac{43}{756}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
| \( \chi_{ 5299 }(1028, a) \) |
\(1\) | \(1\) | \(e\left(\frac{43}{756}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{43}{378}\right)\) | \(e\left(\frac{307}{756}\right)\) | \(e\left(\frac{253}{756}\right)\) | \(e\left(\frac{43}{252}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{25}{54}\right)\) | \(e\left(\frac{44}{189}\right)\) | \(e\left(\frac{74}{189}\right)\) |
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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)
