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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(10023, base_ring=CyclotomicField(256))
M = H._module
chi = DirichletCharacter(H, M([128,0,193]))
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gp:[g,chi] = znchar(Mod(1847, 10023))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("10023.1847");
| Modulus: | \(10023\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(771\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(256\) |
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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: | no, induced from \(\chi_{771}(305,\cdot)\) |
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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);
|
\(\chi_{10023}(14,\cdot)\)
\(\chi_{10023}(53,\cdot)\)
\(\chi_{10023}(131,\cdot)\)
\(\chi_{10023}(170,\cdot)\)
\(\chi_{10023}(209,\cdot)\)
\(\chi_{10023}(326,\cdot)\)
\(\chi_{10023}(365,\cdot)\)
\(\chi_{10023}(404,\cdot)\)
\(\chi_{10023}(443,\cdot)\)
\(\chi_{10023}(521,\cdot)\)
\(\chi_{10023}(599,\cdot)\)
\(\chi_{10023}(677,\cdot)\)
\(\chi_{10023}(716,\cdot)\)
\(\chi_{10023}(872,\cdot)\)
\(\chi_{10023}(950,\cdot)\)
\(\chi_{10023}(989,\cdot)\)
\(\chi_{10023}(1067,\cdot)\)
\(\chi_{10023}(1106,\cdot)\)
\(\chi_{10023}(1184,\cdot)\)
\(\chi_{10023}(1340,\cdot)\)
\(\chi_{10023}(1379,\cdot)\)
\(\chi_{10023}(1457,\cdot)\)
\(\chi_{10023}(1535,\cdot)\)
\(\chi_{10023}(1613,\cdot)\)
\(\chi_{10023}(1652,\cdot)\)
\(\chi_{10023}(1691,\cdot)\)
\(\chi_{10023}(1730,\cdot)\)
\(\chi_{10023}(1847,\cdot)\)
\(\chi_{10023}(1886,\cdot)\)
\(\chi_{10023}(1925,\cdot)\)
...
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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 };
\((6683,7711,1288)\) → \((-1,1,e\left(\frac{193}{256}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(14\) | \(16\) | \(17\) |
| \( \chi_{ 10023 }(1847, a) \) |
\(1\) | \(1\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{247}{256}\right)\) | \(e\left(\frac{21}{256}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{167}{256}\right)\) | \(e\left(\frac{17}{64}\right)\) | \(e\left(\frac{197}{256}\right)\) | \(-i\) | \(e\left(\frac{31}{32}\right)\) |
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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)
