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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4901, base_ring=CyclotomicField(364))
M = H._module
chi = DirichletCharacter(H, M([343,117]))
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gp:[g,chi] = znchar(Mod(541, 4901))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("4901.541");
| Modulus: | \(4901\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(4901\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(364\) |
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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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\(\chi_{4901}(18,\cdot)\)
\(\chi_{4901}(21,\cdot)\)
\(\chi_{4901}(44,\cdot)\)
\(\chi_{4901}(60,\cdot)\)
\(\chi_{4901}(164,\cdot)\)
\(\chi_{4901}(177,\cdot)\)
\(\chi_{4901}(200,\cdot)\)
\(\chi_{4901}(213,\cdot)\)
\(\chi_{4901}(317,\cdot)\)
\(\chi_{4901}(333,\cdot)\)
\(\chi_{4901}(356,\cdot)\)
\(\chi_{4901}(359,\cdot)\)
\(\chi_{4901}(395,\cdot)\)
\(\chi_{4901}(398,\cdot)\)
\(\chi_{4901}(421,\cdot)\)
\(\chi_{4901}(541,\cdot)\)
\(\chi_{4901}(554,\cdot)\)
\(\chi_{4901}(590,\cdot)\)
\(\chi_{4901}(694,\cdot)\)
\(\chi_{4901}(710,\cdot)\)
\(\chi_{4901}(733,\cdot)\)
\(\chi_{4901}(736,\cdot)\)
\(\chi_{4901}(772,\cdot)\)
\(\chi_{4901}(798,\cdot)\)
\(\chi_{4901}(814,\cdot)\)
\(\chi_{4901}(918,\cdot)\)
\(\chi_{4901}(931,\cdot)\)
\(\chi_{4901}(954,\cdot)\)
\(\chi_{4901}(967,\cdot)\)
\(\chi_{4901}(1071,\cdot)\)
...
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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 };
\((1016,3888)\) → \((e\left(\frac{49}{52}\right),e\left(\frac{9}{28}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 4901 }(541, a) \) |
\(1\) | \(1\) | \(e\left(\frac{24}{91}\right)\) | \(e\left(\frac{165}{364}\right)\) | \(e\left(\frac{48}{91}\right)\) | \(e\left(\frac{201}{364}\right)\) | \(e\left(\frac{261}{364}\right)\) | \(e\left(\frac{249}{364}\right)\) | \(e\left(\frac{72}{91}\right)\) | \(e\left(\frac{165}{182}\right)\) | \(e\left(\frac{297}{364}\right)\) | \(e\left(\frac{17}{182}\right)\) |
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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)
