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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2263, base_ring=CyclotomicField(180))
M = H._module
chi = DirichletCharacter(H, M([96,85]))
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gp:[g,chi] = znchar(Mod(400, 2263))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2263.400");
| Modulus: | \(2263\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(2263\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(180\) |
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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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\(\chi_{2263}(38,\cdot)\)
\(\chi_{2263}(50,\cdot)\)
\(\chi_{2263}(165,\cdot)\)
\(\chi_{2263}(169,\cdot)\)
\(\chi_{2263}(231,\cdot)\)
\(\chi_{2263}(359,\cdot)\)
\(\chi_{2263}(390,\cdot)\)
\(\chi_{2263}(400,\cdot)\)
\(\chi_{2263}(413,\cdot)\)
\(\chi_{2263}(444,\cdot)\)
\(\chi_{2263}(505,\cdot)\)
\(\chi_{2263}(534,\cdot)\)
\(\chi_{2263}(536,\cdot)\)
\(\chi_{2263}(546,\cdot)\)
\(\chi_{2263}(572,\cdot)\)
\(\chi_{2263}(578,\cdot)\)
\(\chi_{2263}(596,\cdot)\)
\(\chi_{2263}(609,\cdot)\)
\(\chi_{2263}(634,\cdot)\)
\(\chi_{2263}(638,\cdot)\)
\(\chi_{2263}(784,\cdot)\)
\(\chi_{2263}(857,\cdot)\)
\(\chi_{2263}(937,\cdot)\)
\(\chi_{2263}(999,\cdot)\)
\(\chi_{2263}(1041,\cdot)\)
\(\chi_{2263}(1130,\cdot)\)
\(\chi_{2263}(1187,\cdot)\)
\(\chi_{2263}(1206,\cdot)\)
\(\chi_{2263}(1260,\cdot)\)
\(\chi_{2263}(1289,\cdot)\)
...
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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 };
\((220,1830)\) → \((e\left(\frac{8}{15}\right),e\left(\frac{17}{36}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 2263 }(400, a) \) |
\(1\) | \(1\) | \(e\left(\frac{26}{45}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{7}{45}\right)\) | \(e\left(\frac{5}{36}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{31}{60}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{43}{60}\right)\) | \(e\left(\frac{43}{180}\right)\) |
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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)
