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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3737, base_ring=CyclotomicField(450))
M = H._module
chi = DirichletCharacter(H, M([50,81]))
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gp:[g,chi] = znchar(Mod(756, 3737))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("3737.756");
| Modulus: | \(3737\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(3737\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(450\) |
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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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\(\chi_{3737}(9,\cdot)\)
\(\chi_{3737}(33,\cdot)\)
\(\chi_{3737}(49,\cdot)\)
\(\chi_{3737}(70,\cdot)\)
\(\chi_{3737}(123,\cdot)\)
\(\chi_{3737}(144,\cdot)\)
\(\chi_{3737}(197,\cdot)\)
\(\chi_{3737}(266,\cdot)\)
\(\chi_{3737}(312,\cdot)\)
\(\chi_{3737}(367,\cdot)\)
\(\chi_{3737}(379,\cdot)\)
\(\chi_{3737}(451,\cdot)\)
\(\chi_{3737}(453,\cdot)\)
\(\chi_{3737}(514,\cdot)\)
\(\chi_{3737}(525,\cdot)\)
\(\chi_{3737}(527,\cdot)\)
\(\chi_{3737}(552,\cdot)\)
\(\chi_{3737}(601,\cdot)\)
\(\chi_{3737}(626,\cdot)\)
\(\chi_{3737}(636,\cdot)\)
\(\chi_{3737}(682,\cdot)\)
\(\chi_{3737}(737,\cdot)\)
\(\chi_{3737}(752,\cdot)\)
\(\chi_{3737}(756,\cdot)\)
\(\chi_{3737}(784,\cdot)\)
\(\chi_{3737}(789,\cdot)\)
\(\chi_{3737}(821,\cdot)\)
\(\chi_{3737}(830,\cdot)\)
\(\chi_{3737}(884,\cdot)\)
\(\chi_{3737}(885,\cdot)\)
...
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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 };
\((1112,2628)\) → \((e\left(\frac{1}{9}\right),e\left(\frac{9}{50}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 3737 }(756, a) \) |
\(1\) | \(1\) | \(e\left(\frac{131}{450}\right)\) | \(e\left(\frac{139}{450}\right)\) | \(e\left(\frac{131}{225}\right)\) | \(e\left(\frac{197}{225}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{79}{450}\right)\) | \(e\left(\frac{131}{150}\right)\) | \(e\left(\frac{139}{225}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{101}{150}\right)\) |
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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)
