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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5763, base_ring=CyclotomicField(112))
M = H._module
chi = DirichletCharacter(H, M([56,63,5]))
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gp:[g,chi] = znchar(Mod(1034, 5763))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("5763.1034");
| Modulus: | \(5763\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(5763\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(112\) |
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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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\(\chi_{5763}(29,\cdot)\)
\(\chi_{5763}(260,\cdot)\)
\(\chi_{5763}(428,\cdot)\)
\(\chi_{5763}(584,\cdot)\)
\(\chi_{5763}(632,\cdot)\)
\(\chi_{5763}(770,\cdot)\)
\(\chi_{5763}(887,\cdot)\)
\(\chi_{5763}(941,\cdot)\)
\(\chi_{5763}(980,\cdot)\)
\(\chi_{5763}(1034,\cdot)\)
\(\chi_{5763}(1151,\cdot)\)
\(\chi_{5763}(1289,\cdot)\)
\(\chi_{5763}(1337,\cdot)\)
\(\chi_{5763}(1493,\cdot)\)
\(\chi_{5763}(1661,\cdot)\)
\(\chi_{5763}(1892,\cdot)\)
\(\chi_{5763}(2081,\cdot)\)
\(\chi_{5763}(2114,\cdot)\)
\(\chi_{5763}(2186,\cdot)\)
\(\chi_{5763}(2255,\cdot)\)
\(\chi_{5763}(2315,\cdot)\)
\(\chi_{5763}(2459,\cdot)\)
\(\chi_{5763}(2540,\cdot)\)
\(\chi_{5763}(2642,\cdot)\)
\(\chi_{5763}(2819,\cdot)\)
\(\chi_{5763}(2870,\cdot)\)
\(\chi_{5763}(2900,\cdot)\)
\(\chi_{5763}(2948,\cdot)\)
\(\chi_{5763}(3071,\cdot)\)
\(\chi_{5763}(3167,\cdot)\)
...
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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 };
\((1922,5086,2602)\) → \((-1,e\left(\frac{9}{16}\right),e\left(\frac{5}{112}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 5763 }(1034, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{51}{56}\right)\) | \(e\left(\frac{23}{28}\right)\) | \(e\left(\frac{1}{56}\right)\) | \(e\left(\frac{61}{112}\right)\) | \(e\left(\frac{41}{56}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{83}{112}\right)\) | \(e\left(\frac{13}{56}\right)\) | \(e\left(\frac{51}{112}\right)\) | \(e\left(\frac{9}{14}\right)\) |
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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)
