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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(80465, base_ring=CyclotomicField(660))
M = H._module
chi = DirichletCharacter(H, M([165,440,192,110]))
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gp:[g,chi] = znchar(Mod(14942, 80465))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("80465.14942");
| Modulus: | \(80465\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(80465\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(660\) |
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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);
|
\(\chi_{80465}(312,\cdot)\)
\(\chi_{80465}(1038,\cdot)\)
\(\chi_{80465}(1642,\cdot)\)
\(\chi_{80465}(1908,\cdot)\)
\(\chi_{80465}(2368,\cdot)\)
\(\chi_{80465}(3238,\cdot)\)
\(\chi_{80465}(3903,\cdot)\)
\(\chi_{80465}(4097,\cdot)\)
\(\chi_{80465}(4568,\cdot)\)
\(\chi_{80465}(5427,\cdot)\)
\(\chi_{80465}(6092,\cdot)\)
\(\chi_{80465}(6297,\cdot)\)
\(\chi_{80465}(6757,\cdot)\)
\(\chi_{80465}(7023,\cdot)\)
\(\chi_{80465}(7627,\cdot)\)
\(\chi_{80465}(8292,\cdot)\)
\(\chi_{80465}(8353,\cdot)\)
\(\chi_{80465}(9018,\cdot)\)
\(\chi_{80465}(10553,\cdot)\)
\(\chi_{80465}(11218,\cdot)\)
\(\chi_{80465}(11412,\cdot)\)
\(\chi_{80465}(11883,\cdot)\)
\(\chi_{80465}(12742,\cdot)\)
\(\chi_{80465}(13407,\cdot)\)
\(\chi_{80465}(13612,\cdot)\)
\(\chi_{80465}(14072,\cdot)\)
\(\chi_{80465}(14338,\cdot)\)
\(\chi_{80465}(14942,\cdot)\)
\(\chi_{80465}(15607,\cdot)\)
\(\chi_{80465}(15668,\cdot)\)
...
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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 };
\((32187,22991,79136,38116)\) → \((i,e\left(\frac{2}{3}\right),e\left(\frac{16}{55}\right),e\left(\frac{1}{6}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(8\) | \(9\) | \(12\) | \(13\) | \(16\) | \(17\) |
| \( \chi_{ 80465 }(14942, a) \) |
\(1\) | \(1\) | \(e\left(\frac{9}{220}\right)\) | \(e\left(\frac{11}{60}\right)\) | \(e\left(\frac{9}{110}\right)\) | \(e\left(\frac{37}{165}\right)\) | \(e\left(\frac{27}{220}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{35}{132}\right)\) | \(e\left(\frac{637}{660}\right)\) | \(e\left(\frac{9}{55}\right)\) | \(e\left(\frac{553}{660}\right)\) |
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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)
