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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4153, base_ring=CyclotomicField(346))
M = H._module
chi = DirichletCharacter(H, M([157]))
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gp:[g,chi] = znchar(Mod(79, 4153))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("4153.79");
| Modulus: | \(4153\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(4153\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(346\) |
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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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\(\chi_{4153}(2,\cdot)\)
\(\chi_{4153}(8,\cdot)\)
\(\chi_{4153}(23,\cdot)\)
\(\chi_{4153}(32,\cdot)\)
\(\chi_{4153}(54,\cdot)\)
\(\chi_{4153}(57,\cdot)\)
\(\chi_{4153}(79,\cdot)\)
\(\chi_{4153}(92,\cdot)\)
\(\chi_{4153}(128,\cdot)\)
\(\chi_{4153}(143,\cdot)\)
\(\chi_{4153}(146,\cdot)\)
\(\chi_{4153}(147,\cdot)\)
\(\chi_{4153}(155,\cdot)\)
\(\chi_{4153}(175,\cdot)\)
\(\chi_{4153}(193,\cdot)\)
\(\chi_{4153}(214,\cdot)\)
\(\chi_{4153}(216,\cdot)\)
\(\chi_{4153}(226,\cdot)\)
\(\chi_{4153}(228,\cdot)\)
\(\chi_{4153}(255,\cdot)\)
\(\chi_{4153}(261,\cdot)\)
\(\chi_{4153}(316,\cdot)\)
\(\chi_{4153}(368,\cdot)\)
\(\chi_{4153}(422,\cdot)\)
\(\chi_{4153}(495,\cdot)\)
\(\chi_{4153}(512,\cdot)\)
\(\chi_{4153}(572,\cdot)\)
\(\chi_{4153}(584,\cdot)\)
\(\chi_{4153}(588,\cdot)\)
\(\chi_{4153}(620,\cdot)\)
...
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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 };
\(5\) → \(e\left(\frac{157}{346}\right)\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 4153 }(79, a) \) |
\(1\) | \(1\) | \(e\left(\frac{151}{173}\right)\) | \(e\left(\frac{91}{173}\right)\) | \(e\left(\frac{129}{173}\right)\) | \(e\left(\frac{157}{346}\right)\) | \(e\left(\frac{69}{173}\right)\) | \(e\left(\frac{99}{173}\right)\) | \(e\left(\frac{107}{173}\right)\) | \(e\left(\frac{9}{173}\right)\) | \(e\left(\frac{113}{346}\right)\) | \(e\left(\frac{149}{346}\right)\) |
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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)
