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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4272, base_ring=CyclotomicField(88))
M = H._module
chi = DirichletCharacter(H, M([0,66,44,51]))
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gp:[g,chi] = znchar(Mod(1373, 4272))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("4272.1373");
| Modulus: | \(4272\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(4272\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(88\) |
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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: | even |
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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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\(\chi_{4272}(29,\cdot)\)
\(\chi_{4272}(389,\cdot)\)
\(\chi_{4272}(629,\cdot)\)
\(\chi_{4272}(653,\cdot)\)
\(\chi_{4272}(773,\cdot)\)
\(\chi_{4272}(941,\cdot)\)
\(\chi_{4272}(965,\cdot)\)
\(\chi_{4272}(1037,\cdot)\)
\(\chi_{4272}(1061,\cdot)\)
\(\chi_{4272}(1109,\cdot)\)
\(\chi_{4272}(1205,\cdot)\)
\(\chi_{4272}(1253,\cdot)\)
\(\chi_{4272}(1277,\cdot)\)
\(\chi_{4272}(1349,\cdot)\)
\(\chi_{4272}(1373,\cdot)\)
\(\chi_{4272}(1541,\cdot)\)
\(\chi_{4272}(1661,\cdot)\)
\(\chi_{4272}(1685,\cdot)\)
\(\chi_{4272}(1925,\cdot)\)
\(\chi_{4272}(2285,\cdot)\)
\(\chi_{4272}(2333,\cdot)\)
\(\chi_{4272}(2357,\cdot)\)
\(\chi_{4272}(2429,\cdot)\)
\(\chi_{4272}(2477,\cdot)\)
\(\chi_{4272}(2693,\cdot)\)
\(\chi_{4272}(2813,\cdot)\)
\(\chi_{4272}(2861,\cdot)\)
\(\chi_{4272}(3029,\cdot)\)
\(\chi_{4272}(3053,\cdot)\)
\(\chi_{4272}(3269,\cdot)\)
...
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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 };
\((2671,3205,2849,1249)\) → \((1,-i,-1,e\left(\frac{51}{88}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
| \( \chi_{ 4272 }(1373, a) \) |
\(1\) | \(1\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{39}{88}\right)\) | \(e\left(\frac{41}{44}\right)\) | \(e\left(\frac{51}{88}\right)\) | \(e\left(\frac{43}{44}\right)\) | \(e\left(\frac{47}{88}\right)\) | \(e\left(\frac{3}{88}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{83}{88}\right)\) | \(e\left(\frac{85}{88}\right)\) |
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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)
