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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(88765, base_ring=CyclotomicField(432))
M = H._module
chi = DirichletCharacter(H, M([0,0,391]))
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gp:[g,chi] = znchar(Mod(1026, 88765))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("88765.1026");
| Modulus: | \(88765\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(433\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(432\) |
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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 \(\chi_{433}(160,\cdot)\) |
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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: | 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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\(\chi_{88765}(1026,\cdot)\)
\(\chi_{88765}(1846,\cdot)\)
\(\chi_{88765}(2051,\cdot)\)
\(\chi_{88765}(2871,\cdot)\)
\(\chi_{88765}(4511,\cdot)\)
\(\chi_{88765}(4716,\cdot)\)
\(\chi_{88765}(5331,\cdot)\)
\(\chi_{88765}(5536,\cdot)\)
\(\chi_{88765}(5741,\cdot)\)
\(\chi_{88765}(5946,\cdot)\)
\(\chi_{88765}(7176,\cdot)\)
\(\chi_{88765}(7381,\cdot)\)
\(\chi_{88765}(7996,\cdot)\)
\(\chi_{88765}(9636,\cdot)\)
\(\chi_{88765}(10046,\cdot)\)
\(\chi_{88765}(11686,\cdot)\)
\(\chi_{88765}(12096,\cdot)\)
\(\chi_{88765}(13736,\cdot)\)
\(\chi_{88765}(13941,\cdot)\)
\(\chi_{88765}(14351,\cdot)\)
\(\chi_{88765}(14556,\cdot)\)
\(\chi_{88765}(14966,\cdot)\)
\(\chi_{88765}(15581,\cdot)\)
\(\chi_{88765}(15991,\cdot)\)
\(\chi_{88765}(16196,\cdot)\)
\(\chi_{88765}(16606,\cdot)\)
\(\chi_{88765}(16811,\cdot)\)
\(\chi_{88765}(17016,\cdot)\)
\(\chi_{88765}(17631,\cdot)\)
\(\chi_{88765}(17836,\cdot)\)
...
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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 };
\((35507,54126,80976)\) → \((1,1,e\left(\frac{391}{432}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
| \( \chi_{ 88765 }(1026, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{43}{72}\right)\) | \(e\left(\frac{1}{27}\right)\) | \(e\left(\frac{7}{36}\right)\) | \(e\left(\frac{137}{216}\right)\) | \(e\left(\frac{155}{432}\right)\) | \(e\left(\frac{19}{24}\right)\) | \(e\left(\frac{2}{27}\right)\) | \(e\left(\frac{143}{216}\right)\) | \(e\left(\frac{25}{108}\right)\) | \(e\left(\frac{215}{216}\right)\) |
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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)
