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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(90900, base_ring=CyclotomicField(300))
M = H._module
chi = DirichletCharacter(H, M([150,250,105,123]))
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gp:[g,chi] = znchar(Mod(33503, 90900))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("90900.33503");
| Modulus: | \(90900\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(90900\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(300\) |
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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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\(\chi_{90900}(83,\cdot)\)
\(\chi_{90900}(263,\cdot)\)
\(\chi_{90900}(1667,\cdot)\)
\(\chi_{90900}(1847,\cdot)\)
\(\chi_{90900}(2567,\cdot)\)
\(\chi_{90900}(3827,\cdot)\)
\(\chi_{90900}(4187,\cdot)\)
\(\chi_{90900}(7067,\cdot)\)
\(\chi_{90900}(11603,\cdot)\)
\(\chi_{90900}(12047,\cdot)\)
\(\chi_{90900}(12587,\cdot)\)
\(\chi_{90900}(13223,\cdot)\)
\(\chi_{90900}(16187,\cdot)\)
\(\chi_{90900}(17627,\cdot)\)
\(\chi_{90900}(17723,\cdot)\)
\(\chi_{90900}(19163,\cdot)\)
\(\chi_{90900}(22127,\cdot)\)
\(\chi_{90900}(22763,\cdot)\)
\(\chi_{90900}(23303,\cdot)\)
\(\chi_{90900}(23747,\cdot)\)
\(\chi_{90900}(28283,\cdot)\)
\(\chi_{90900}(31163,\cdot)\)
\(\chi_{90900}(31523,\cdot)\)
\(\chi_{90900}(32783,\cdot)\)
\(\chi_{90900}(33503,\cdot)\)
\(\chi_{90900}(33683,\cdot)\)
\(\chi_{90900}(35087,\cdot)\)
\(\chi_{90900}(35267,\cdot)\)
\(\chi_{90900}(36527,\cdot)\)
\(\chi_{90900}(39263,\cdot)\)
...
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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 };
\((45451,50501,58177,50401)\) → \((-1,e\left(\frac{5}{6}\right),e\left(\frac{7}{20}\right),e\left(\frac{41}{100}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 90900 }(33503, a) \) |
\(1\) | \(1\) | \(e\left(\frac{41}{150}\right)\) | \(e\left(\frac{79}{300}\right)\) | \(e\left(\frac{113}{300}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{4}{25}\right)\) | \(e\left(\frac{233}{300}\right)\) | \(e\left(\frac{253}{300}\right)\) | \(e\left(\frac{61}{150}\right)\) | \(e\left(\frac{11}{100}\right)\) | \(e\left(\frac{1}{60}\right)\) |
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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)
