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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(164560, base_ring=CyclotomicField(110))
M = H._module
chi = DirichletCharacter(H, M([0,0,0,57,0]))
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gp:[g,chi] = znchar(Mod(5441, 164560))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("164560.5441");
| Modulus: | \(164560\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(121\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(110\) |
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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: | no, induced from \(\chi_{121}(117,\cdot)\) |
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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: | no |
| 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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\(\chi_{164560}(1361,\cdot)\)
\(\chi_{164560}(5441,\cdot)\)
\(\chi_{164560}(9521,\cdot)\)
\(\chi_{164560}(16321,\cdot)\)
\(\chi_{164560}(20401,\cdot)\)
\(\chi_{164560}(24481,\cdot)\)
\(\chi_{164560}(25841,\cdot)\)
\(\chi_{164560}(31281,\cdot)\)
\(\chi_{164560}(35361,\cdot)\)
\(\chi_{164560}(39441,\cdot)\)
\(\chi_{164560}(40801,\cdot)\)
\(\chi_{164560}(46241,\cdot)\)
\(\chi_{164560}(50321,\cdot)\)
\(\chi_{164560}(54401,\cdot)\)
\(\chi_{164560}(55761,\cdot)\)
\(\chi_{164560}(61201,\cdot)\)
\(\chi_{164560}(65281,\cdot)\)
\(\chi_{164560}(69361,\cdot)\)
\(\chi_{164560}(70721,\cdot)\)
\(\chi_{164560}(76161,\cdot)\)
\(\chi_{164560}(80241,\cdot)\)
\(\chi_{164560}(84321,\cdot)\)
\(\chi_{164560}(85681,\cdot)\)
\(\chi_{164560}(91121,\cdot)\)
\(\chi_{164560}(95201,\cdot)\)
\(\chi_{164560}(99281,\cdot)\)
\(\chi_{164560}(100641,\cdot)\)
\(\chi_{164560}(106081,\cdot)\)
\(\chi_{164560}(110161,\cdot)\)
\(\chi_{164560}(114241,\cdot)\)
...
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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 };
\((61711,41141,98737,130561,96801)\) → \((1,1,1,e\left(\frac{57}{110}\right),1)\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(13\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) | \(31\) |
| \( \chi_{ 164560 }(5441, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{69}{110}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{37}{110}\right)\) | \(e\left(\frac{1}{110}\right)\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{89}{110}\right)\) | \(e\left(\frac{31}{55}\right)\) |
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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)
