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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5004, base_ring=CyclotomicField(138))
M = H._module
chi = DirichletCharacter(H, M([0,0,49]))
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gp:[g,chi] = znchar(Mod(73, 5004))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("5004.73");
| Modulus: | \(5004\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(139\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(138\) |
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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 \(\chi_{139}(73,\cdot)\) |
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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: | 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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\(\chi_{5004}(73,\cdot)\)
\(\chi_{5004}(109,\cdot)\)
\(\chi_{5004}(253,\cdot)\)
\(\chi_{5004}(397,\cdot)\)
\(\chi_{5004}(505,\cdot)\)
\(\chi_{5004}(577,\cdot)\)
\(\chi_{5004}(649,\cdot)\)
\(\chi_{5004}(721,\cdot)\)
\(\chi_{5004}(793,\cdot)\)
\(\chi_{5004}(829,\cdot)\)
\(\chi_{5004}(1045,\cdot)\)
\(\chi_{5004}(1081,\cdot)\)
\(\chi_{5004}(1405,\cdot)\)
\(\chi_{5004}(1513,\cdot)\)
\(\chi_{5004}(1585,\cdot)\)
\(\chi_{5004}(1621,\cdot)\)
\(\chi_{5004}(1657,\cdot)\)
\(\chi_{5004}(1729,\cdot)\)
\(\chi_{5004}(1909,\cdot)\)
\(\chi_{5004}(2125,\cdot)\)
\(\chi_{5004}(2413,\cdot)\)
\(\chi_{5004}(2521,\cdot)\)
\(\chi_{5004}(2773,\cdot)\)
\(\chi_{5004}(2881,\cdot)\)
\(\chi_{5004}(2989,\cdot)\)
\(\chi_{5004}(3061,\cdot)\)
\(\chi_{5004}(3169,\cdot)\)
\(\chi_{5004}(3421,\cdot)\)
\(\chi_{5004}(3493,\cdot)\)
\(\chi_{5004}(3565,\cdot)\)
...
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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 };
\((2503,2225,4033)\) → \((1,1,e\left(\frac{49}{138}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
| \( \chi_{ 5004 }(73, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{37}{69}\right)\) | \(e\left(\frac{52}{69}\right)\) | \(e\left(\frac{68}{69}\right)\) | \(e\left(\frac{50}{69}\right)\) | \(e\left(\frac{137}{138}\right)\) | \(e\left(\frac{91}{138}\right)\) | \(e\left(\frac{27}{46}\right)\) | \(e\left(\frac{5}{69}\right)\) | \(e\left(\frac{26}{69}\right)\) | \(e\left(\frac{61}{69}\right)\) |
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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)
