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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(27797, base_ring=CyclotomicField(570))
M = H._module
chi = DirichletCharacter(H, M([190,342,335]))
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gp:[g,chi] = znchar(Mod(2858, 27797))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("27797.2858");
| Modulus: | \(27797\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(27797\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(570\) |
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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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\(\chi_{27797}(487,\cdot)\)
\(\chi_{27797}(597,\cdot)\)
\(\chi_{27797}(620,\cdot)\)
\(\chi_{27797}(730,\cdot)\)
\(\chi_{27797}(753,\cdot)\)
\(\chi_{27797}(863,\cdot)\)
\(\chi_{27797}(1285,\cdot)\)
\(\chi_{27797}(1395,\cdot)\)
\(\chi_{27797}(1950,\cdot)\)
\(\chi_{27797}(2060,\cdot)\)
\(\chi_{27797}(2083,\cdot)\)
\(\chi_{27797}(2193,\cdot)\)
\(\chi_{27797}(2216,\cdot)\)
\(\chi_{27797}(2326,\cdot)\)
\(\chi_{27797}(2748,\cdot)\)
\(\chi_{27797}(2858,\cdot)\)
\(\chi_{27797}(3413,\cdot)\)
\(\chi_{27797}(3523,\cdot)\)
\(\chi_{27797}(3546,\cdot)\)
\(\chi_{27797}(3656,\cdot)\)
\(\chi_{27797}(3789,\cdot)\)
\(\chi_{27797}(4211,\cdot)\)
\(\chi_{27797}(4321,\cdot)\)
\(\chi_{27797}(4876,\cdot)\)
\(\chi_{27797}(5009,\cdot)\)
\(\chi_{27797}(5119,\cdot)\)
\(\chi_{27797}(5142,\cdot)\)
\(\chi_{27797}(5252,\cdot)\)
\(\chi_{27797}(5674,\cdot)\)
\(\chi_{27797}(5784,\cdot)\)
...
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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 };
\((3972,17690,22023)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{3}{5}\right),e\left(\frac{67}{114}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(12\) | \(13\) |
| \( \chi_{ 27797 }(2858, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{487}{570}\right)\) | \(e\left(\frac{157}{190}\right)\) | \(e\left(\frac{202}{285}\right)\) | \(e\left(\frac{119}{285}\right)\) | \(e\left(\frac{194}{285}\right)\) | \(e\left(\frac{107}{190}\right)\) | \(e\left(\frac{62}{95}\right)\) | \(e\left(\frac{31}{114}\right)\) | \(e\left(\frac{61}{114}\right)\) | \(e\left(\frac{547}{570}\right)\) |
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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)
