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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8320, base_ring=CyclotomicField(96))
M = H._module
chi = DirichletCharacter(H, M([48,3,24,56]))
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gp:[g,chi] = znchar(Mod(5627, 8320))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("8320.5627");
| Modulus: | \(8320\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(8320\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(96\) |
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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: | odd |
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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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\(\chi_{8320}(267,\cdot)\)
\(\chi_{8320}(427,\cdot)\)
\(\chi_{8320}(483,\cdot)\)
\(\chi_{8320}(643,\cdot)\)
\(\chi_{8320}(1307,\cdot)\)
\(\chi_{8320}(1467,\cdot)\)
\(\chi_{8320}(1523,\cdot)\)
\(\chi_{8320}(1683,\cdot)\)
\(\chi_{8320}(2347,\cdot)\)
\(\chi_{8320}(2507,\cdot)\)
\(\chi_{8320}(2563,\cdot)\)
\(\chi_{8320}(2723,\cdot)\)
\(\chi_{8320}(3387,\cdot)\)
\(\chi_{8320}(3547,\cdot)\)
\(\chi_{8320}(3603,\cdot)\)
\(\chi_{8320}(3763,\cdot)\)
\(\chi_{8320}(4427,\cdot)\)
\(\chi_{8320}(4587,\cdot)\)
\(\chi_{8320}(4643,\cdot)\)
\(\chi_{8320}(4803,\cdot)\)
\(\chi_{8320}(5467,\cdot)\)
\(\chi_{8320}(5627,\cdot)\)
\(\chi_{8320}(5683,\cdot)\)
\(\chi_{8320}(5843,\cdot)\)
\(\chi_{8320}(6507,\cdot)\)
\(\chi_{8320}(6667,\cdot)\)
\(\chi_{8320}(6723,\cdot)\)
\(\chi_{8320}(6883,\cdot)\)
\(\chi_{8320}(7547,\cdot)\)
\(\chi_{8320}(7707,\cdot)\)
...
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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 };
\((8191,261,6657,5761)\) → \((-1,e\left(\frac{1}{32}\right),i,e\left(\frac{7}{12}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 8320 }(5627, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{65}{96}\right)\) | \(e\left(\frac{23}{48}\right)\) | \(e\left(\frac{17}{48}\right)\) | \(e\left(\frac{23}{96}\right)\) | \(e\left(\frac{7}{24}\right)\) | \(e\left(\frac{61}{96}\right)\) | \(e\left(\frac{5}{32}\right)\) | \(e\left(\frac{25}{48}\right)\) | \(e\left(\frac{1}{32}\right)\) | \(e\left(\frac{65}{96}\right)\) |
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sage:chi(x) # x integer
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gp:chareval(g,chi,x) \\ x integer, value in Q/Z
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magma:chi(x)
