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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(47311, base_ring=CyclotomicField(88))
M = H._module
chi = DirichletCharacter(H, M([68,77,12]))
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gp:[g,chi] = znchar(Mod(18226, 47311))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("47311.18226");
| Modulus: | \(47311\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(47311\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(88\) |
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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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\(\chi_{47311}(1528,\cdot)\)
\(\chi_{47311}(2100,\cdot)\)
\(\chi_{47311}(6621,\cdot)\)
\(\chi_{47311}(8117,\cdot)\)
\(\chi_{47311}(8502,\cdot)\)
\(\chi_{47311}(9987,\cdot)\)
\(\chi_{47311}(10185,\cdot)\)
\(\chi_{47311}(11120,\cdot)\)
\(\chi_{47311}(11307,\cdot)\)
\(\chi_{47311}(12187,\cdot)\)
\(\chi_{47311}(13683,\cdot)\)
\(\chi_{47311}(14068,\cdot)\)
\(\chi_{47311}(15553,\cdot)\)
\(\chi_{47311}(15751,\cdot)\)
\(\chi_{47311}(16686,\cdot)\)
\(\chi_{47311}(16873,\cdot)\)
\(\chi_{47311}(18226,\cdot)\)
\(\chi_{47311}(19337,\cdot)\)
\(\chi_{47311}(21581,\cdot)\)
\(\chi_{47311}(23792,\cdot)\)
\(\chi_{47311}(24903,\cdot)\)
\(\chi_{47311}(27147,\cdot)\)
\(\chi_{47311}(28885,\cdot)\)
\(\chi_{47311}(30381,\cdot)\)
\(\chi_{47311}(32251,\cdot)\)
\(\chi_{47311}(33549,\cdot)\)
\(\chi_{47311}(34451,\cdot)\)
\(\chi_{47311}(35232,\cdot)\)
\(\chi_{47311}(35947,\cdot)\)
\(\chi_{47311}(36167,\cdot)\)
...
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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 };
\((5084,8350,10286)\) → \((e\left(\frac{17}{22}\right),e\left(\frac{7}{8}\right),e\left(\frac{3}{22}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(12\) |
| \( \chi_{ 47311 }(18226, a) \) |
\(1\) | \(1\) | \(e\left(\frac{13}{44}\right)\) | \(e\left(\frac{5}{88}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{61}{88}\right)\) | \(e\left(\frac{31}{88}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{39}{44}\right)\) | \(e\left(\frac{5}{44}\right)\) | \(e\left(\frac{87}{88}\right)\) | \(e\left(\frac{57}{88}\right)\) |
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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)
