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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(80586, base_ring=CyclotomicField(990))
M = H._module
chi = DirichletCharacter(H, M([660,306,605]))
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gp:[g,chi] = znchar(Mod(8827, 80586))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("80586.8827");
| Modulus: | \(80586\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(40293\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(990\) |
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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 \(\chi_{40293}(8827,\cdot)\) |
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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: | 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);
|
\(\chi_{80586}(25,\cdot)\)
\(\chi_{80586}(169,\cdot)\)
\(\chi_{80586}(691,\cdot)\)
\(\chi_{80586}(1039,\cdot)\)
\(\chi_{80586}(1357,\cdot)\)
\(\chi_{80586}(1501,\cdot)\)
\(\chi_{80586}(2335,\cdot)\)
\(\chi_{80586}(2731,\cdot)\)
\(\chi_{80586}(3001,\cdot)\)
\(\chi_{80586}(3667,\cdot)\)
\(\chi_{80586}(4063,\cdot)\)
\(\chi_{80586}(4579,\cdot)\)
\(\chi_{80586}(4999,\cdot)\)
\(\chi_{80586}(5245,\cdot)\)
\(\chi_{80586}(5395,\cdot)\)
\(\chi_{80586}(5701,\cdot)\)
\(\chi_{80586}(5911,\cdot)\)
\(\chi_{80586}(6163,\cdot)\)
\(\chi_{80586}(6367,\cdot)\)
\(\chi_{80586}(6829,\cdot)\)
\(\chi_{80586}(7033,\cdot)\)
\(\chi_{80586}(7243,\cdot)\)
\(\chi_{80586}(7351,\cdot)\)
\(\chi_{80586}(7495,\cdot)\)
\(\chi_{80586}(8017,\cdot)\)
\(\chi_{80586}(8365,\cdot)\)
\(\chi_{80586}(8683,\cdot)\)
\(\chi_{80586}(8827,\cdot)\)
\(\chi_{80586}(9661,\cdot)\)
\(\chi_{80586}(10015,\cdot)\)
...
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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 };
\((71633,1333,47917)\) → \((e\left(\frac{2}{3}\right),e\left(\frac{17}{55}\right),e\left(\frac{11}{18}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 80586 }(8827, a) \) |
\(1\) | \(1\) | \(e\left(\frac{259}{990}\right)\) | \(e\left(\frac{191}{495}\right)\) | \(e\left(\frac{271}{990}\right)\) | \(e\left(\frac{419}{990}\right)\) | \(e\left(\frac{43}{990}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{259}{495}\right)\) | \(e\left(\frac{83}{110}\right)\) | \(e\left(\frac{137}{330}\right)\) | \(e\left(\frac{641}{990}\right)\) |
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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)
