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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3787, base_ring=CyclotomicField(540))
M = H._module
chi = DirichletCharacter(H, M([450,499]))
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gp:[g,chi] = znchar(Mod(40, 3787))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("3787.40");
| Modulus: | \(3787\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(3787\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(540\) |
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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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\(\chi_{3787}(10,\cdot)\)
\(\chi_{3787}(24,\cdot)\)
\(\chi_{3787}(40,\cdot)\)
\(\chi_{3787}(54,\cdot)\)
\(\chi_{3787}(87,\cdot)\)
\(\chi_{3787}(94,\cdot)\)
\(\chi_{3787}(96,\cdot)\)
\(\chi_{3787}(131,\cdot)\)
\(\chi_{3787}(150,\cdot)\)
\(\chi_{3787}(206,\cdot)\)
\(\chi_{3787}(213,\cdot)\)
\(\chi_{3787}(250,\cdot)\)
\(\chi_{3787}(318,\cdot)\)
\(\chi_{3787}(360,\cdot)\)
\(\chi_{3787}(409,\cdot)\)
\(\chi_{3787}(474,\cdot)\)
\(\chi_{3787}(479,\cdot)\)
\(\chi_{3787}(486,\cdot)\)
\(\chi_{3787}(600,\cdot)\)
\(\chi_{3787}(614,\cdot)\)
\(\chi_{3787}(640,\cdot)\)
\(\chi_{3787}(717,\cdot)\)
\(\chi_{3787}(740,\cdot)\)
\(\chi_{3787}(759,\cdot)\)
\(\chi_{3787}(761,\cdot)\)
\(\chi_{3787}(789,\cdot)\)
\(\chi_{3787}(808,\cdot)\)
\(\chi_{3787}(810,\cdot)\)
\(\chi_{3787}(815,\cdot)\)
\(\chi_{3787}(824,\cdot)\)
...
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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 };
\((542,1625)\) → \((e\left(\frac{5}{6}\right),e\left(\frac{499}{540}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
| \( \chi_{ 3787 }(40, a) \) |
\(1\) | \(1\) | \(e\left(\frac{319}{540}\right)\) | \(e\left(\frac{253}{270}\right)\) | \(e\left(\frac{49}{270}\right)\) | \(e\left(\frac{137}{270}\right)\) | \(e\left(\frac{19}{36}\right)\) | \(e\left(\frac{139}{180}\right)\) | \(e\left(\frac{118}{135}\right)\) | \(e\left(\frac{53}{540}\right)\) | \(e\left(\frac{101}{108}\right)\) | \(e\left(\frac{16}{135}\right)\) |
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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)
