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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4624, base_ring=CyclotomicField(136))
M = H._module
chi = DirichletCharacter(H, M([68,34,81]))
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gp:[g,chi] = znchar(Mod(859, 4624))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("4624.859");
| Modulus: | \(4624\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(4624\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(136\) |
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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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\(\chi_{4624}(19,\cdot)\)
\(\chi_{4624}(43,\cdot)\)
\(\chi_{4624}(59,\cdot)\)
\(\chi_{4624}(83,\cdot)\)
\(\chi_{4624}(291,\cdot)\)
\(\chi_{4624}(315,\cdot)\)
\(\chi_{4624}(331,\cdot)\)
\(\chi_{4624}(355,\cdot)\)
\(\chi_{4624}(563,\cdot)\)
\(\chi_{4624}(587,\cdot)\)
\(\chi_{4624}(603,\cdot)\)
\(\chi_{4624}(627,\cdot)\)
\(\chi_{4624}(835,\cdot)\)
\(\chi_{4624}(859,\cdot)\)
\(\chi_{4624}(875,\cdot)\)
\(\chi_{4624}(899,\cdot)\)
\(\chi_{4624}(1107,\cdot)\)
\(\chi_{4624}(1131,\cdot)\)
\(\chi_{4624}(1147,\cdot)\)
\(\chi_{4624}(1171,\cdot)\)
\(\chi_{4624}(1379,\cdot)\)
\(\chi_{4624}(1403,\cdot)\)
\(\chi_{4624}(1419,\cdot)\)
\(\chi_{4624}(1443,\cdot)\)
\(\chi_{4624}(1651,\cdot)\)
\(\chi_{4624}(1675,\cdot)\)
\(\chi_{4624}(1691,\cdot)\)
\(\chi_{4624}(1715,\cdot)\)
\(\chi_{4624}(1923,\cdot)\)
\(\chi_{4624}(1947,\cdot)\)
...
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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 };
\((4047,1157,4049)\) → \((-1,i,e\left(\frac{81}{136}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(19\) | \(21\) | \(23\) |
| \( \chi_{ 4624 }(859, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{115}{136}\right)\) | \(e\left(\frac{87}{136}\right)\) | \(e\left(\frac{43}{136}\right)\) | \(e\left(\frac{47}{68}\right)\) | \(e\left(\frac{61}{136}\right)\) | \(e\left(\frac{33}{68}\right)\) | \(e\left(\frac{33}{68}\right)\) | \(e\left(\frac{10}{17}\right)\) | \(e\left(\frac{11}{68}\right)\) | \(e\left(\frac{111}{136}\right)\) |
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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)
