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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1477, base_ring=CyclotomicField(210))
M = H._module
chi = DirichletCharacter(H, M([140,17]))
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gp:[g,chi] = znchar(Mod(1096, 1477))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1477.1096");
| Modulus: | \(1477\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(1477\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(210\) |
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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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\(\chi_{1477}(2,\cdot)\)
\(\chi_{1477}(142,\cdot)\)
\(\chi_{1477}(207,\cdot)\)
\(\chi_{1477}(233,\cdot)\)
\(\chi_{1477}(296,\cdot)\)
\(\chi_{1477}(319,\cdot)\)
\(\chi_{1477}(338,\cdot)\)
\(\chi_{1477}(366,\cdot)\)
\(\chi_{1477}(429,\cdot)\)
\(\chi_{1477}(457,\cdot)\)
\(\chi_{1477}(494,\cdot)\)
\(\chi_{1477}(513,\cdot)\)
\(\chi_{1477}(571,\cdot)\)
\(\chi_{1477}(597,\cdot)\)
\(\chi_{1477}(613,\cdot)\)
\(\chi_{1477}(627,\cdot)\)
\(\chi_{1477}(662,\cdot)\)
\(\chi_{1477}(690,\cdot)\)
\(\chi_{1477}(725,\cdot)\)
\(\chi_{1477}(739,\cdot)\)
\(\chi_{1477}(774,\cdot)\)
\(\chi_{1477}(795,\cdot)\)
\(\chi_{1477}(807,\cdot)\)
\(\chi_{1477}(828,\cdot)\)
\(\chi_{1477}(919,\cdot)\)
\(\chi_{1477}(956,\cdot)\)
\(\chi_{1477}(975,\cdot)\)
\(\chi_{1477}(977,\cdot)\)
\(\chi_{1477}(996,\cdot)\)
\(\chi_{1477}(1031,\cdot)\)
...
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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 };
\((423,1268)\) → \((e\left(\frac{2}{3}\right),e\left(\frac{17}{210}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
| \( \chi_{ 1477 }(1096, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{29}{70}\right)\) | \(e\left(\frac{31}{210}\right)\) | \(e\left(\frac{29}{35}\right)\) | \(e\left(\frac{2}{105}\right)\) | \(e\left(\frac{59}{105}\right)\) | \(e\left(\frac{17}{70}\right)\) | \(e\left(\frac{31}{105}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{82}{105}\right)\) | \(e\left(\frac{41}{42}\right)\) |
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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)
