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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(36784, base_ring=CyclotomicField(220))
M = H._module
chi = DirichletCharacter(H, M([0,165,4,110]))
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gp:[g,chi] = znchar(Mod(11741, 36784))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("36784.11741");
| Modulus: | \(36784\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(36784\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(220\) |
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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: | 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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\(\chi_{36784}(37,\cdot)\)
\(\chi_{36784}(797,\cdot)\)
\(\chi_{36784}(949,\cdot)\)
\(\chi_{36784}(1709,\cdot)\)
\(\chi_{36784}(2165,\cdot)\)
\(\chi_{36784}(2469,\cdot)\)
\(\chi_{36784}(2621,\cdot)\)
\(\chi_{36784}(3381,\cdot)\)
\(\chi_{36784}(3837,\cdot)\)
\(\chi_{36784}(4293,\cdot)\)
\(\chi_{36784}(5053,\cdot)\)
\(\chi_{36784}(5509,\cdot)\)
\(\chi_{36784}(5813,\cdot)\)
\(\chi_{36784}(5965,\cdot)\)
\(\chi_{36784}(6725,\cdot)\)
\(\chi_{36784}(7181,\cdot)\)
\(\chi_{36784}(7485,\cdot)\)
\(\chi_{36784}(7637,\cdot)\)
\(\chi_{36784}(8397,\cdot)\)
\(\chi_{36784}(8853,\cdot)\)
\(\chi_{36784}(9157,\cdot)\)
\(\chi_{36784}(9309,\cdot)\)
\(\chi_{36784}(10069,\cdot)\)
\(\chi_{36784}(10525,\cdot)\)
\(\chi_{36784}(10829,\cdot)\)
\(\chi_{36784}(10981,\cdot)\)
\(\chi_{36784}(11741,\cdot)\)
\(\chi_{36784}(12197,\cdot)\)
\(\chi_{36784}(12501,\cdot)\)
\(\chi_{36784}(12653,\cdot)\)
...
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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 };
\((22991,27589,12465,17425)\) → \((1,-i,e\left(\frac{1}{55}\right),-1)\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(13\) | \(15\) | \(17\) | \(21\) | \(23\) | \(25\) |
| \( \chi_{ 36784 }(11741, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{21}{220}\right)\) | \(e\left(\frac{69}{110}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{129}{220}\right)\) | \(e\left(\frac{49}{110}\right)\) | \(e\left(\frac{49}{55}\right)\) | \(e\left(\frac{43}{44}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{21}{110}\right)\) |
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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)
