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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(46651, base_ring=CyclotomicField(4240))
M = H._module
chi = DirichletCharacter(H, M([424,3401]))
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gp:[g,chi] = znchar(Mod(233, 46651))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("46651.233");
| Modulus: | \(46651\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(46651\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(4240\) |
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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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\(\chi_{46651}(6,\cdot)\)
\(\chi_{46651}(30,\cdot)\)
\(\chi_{46651}(57,\cdot)\)
\(\chi_{46651}(74,\cdot)\)
\(\chi_{46651}(94,\cdot)\)
\(\chi_{46651}(134,\cdot)\)
\(\chi_{46651}(150,\cdot)\)
\(\chi_{46651}(204,\cdot)\)
\(\chi_{46651}(215,\cdot)\)
\(\chi_{46651}(216,\cdot)\)
\(\chi_{46651}(233,\cdot)\)
\(\chi_{46651}(244,\cdot)\)
\(\chi_{46651}(316,\cdot)\)
\(\chi_{46651}(370,\cdot)\)
\(\chi_{46651}(448,\cdot)\)
\(\chi_{46651}(470,\cdot)\)
\(\chi_{46651}(503,\cdot)\)
\(\chi_{46651}(547,\cdot)\)
\(\chi_{46651}(567,\cdot)\)
\(\chi_{46651}(574,\cdot)\)
\(\chi_{46651}(618,\cdot)\)
\(\chi_{46651}(712,\cdot)\)
\(\chi_{46651}(723,\cdot)\)
\(\chi_{46651}(728,\cdot)\)
\(\chi_{46651}(744,\cdot)\)
\(\chi_{46651}(750,\cdot)\)
\(\chi_{46651}(761,\cdot)\)
\(\chi_{46651}(831,\cdot)\)
\(\chi_{46651}(893,\cdot)\)
\(\chi_{46651}(981,\cdot)\)
...
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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 };
\((8483,29690)\) → \((e\left(\frac{1}{10}\right),e\left(\frac{3401}{4240}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(12\) |
| \( \chi_{ 46651 }(233, a) \) |
\(1\) | \(1\) | \(e\left(\frac{1261}{2120}\right)\) | \(e\left(\frac{2553}{4240}\right)\) | \(e\left(\frac{201}{1060}\right)\) | \(e\left(\frac{77}{424}\right)\) | \(e\left(\frac{167}{848}\right)\) | \(e\left(\frac{3043}{4240}\right)\) | \(e\left(\frac{1663}{2120}\right)\) | \(e\left(\frac{433}{2120}\right)\) | \(e\left(\frac{823}{1060}\right)\) | \(e\left(\frac{3357}{4240}\right)\) |
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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)
