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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(26825, base_ring=CyclotomicField(180))
M = H._module
chi = DirichletCharacter(H, M([9,45,25]))
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gp:[g,chi] = znchar(Mod(13352, 26825))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("26825.13352");
| Modulus: | \(26825\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(26825\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(180\) |
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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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\(\chi_{26825}(133,\cdot)\)
\(\chi_{26825}(278,\cdot)\)
\(\chi_{26825}(2163,\cdot)\)
\(\chi_{26825}(2622,\cdot)\)
\(\chi_{26825}(3202,\cdot)\)
\(\chi_{26825}(4072,\cdot)\)
\(\chi_{26825}(4797,\cdot)\)
\(\chi_{26825}(5087,\cdot)\)
\(\chi_{26825}(5498,\cdot)\)
\(\chi_{26825}(5933,\cdot)\)
\(\chi_{26825}(6658,\cdot)\)
\(\chi_{26825}(7528,\cdot)\)
\(\chi_{26825}(7987,\cdot)\)
\(\chi_{26825}(8108,\cdot)\)
\(\chi_{26825}(8567,\cdot)\)
\(\chi_{26825}(9437,\cdot)\)
\(\chi_{26825}(10162,\cdot)\)
\(\chi_{26825}(10452,\cdot)\)
\(\chi_{26825}(10597,\cdot)\)
\(\chi_{26825}(10863,\cdot)\)
\(\chi_{26825}(11008,\cdot)\)
\(\chi_{26825}(11298,\cdot)\)
\(\chi_{26825}(12023,\cdot)\)
\(\chi_{26825}(13352,\cdot)\)
\(\chi_{26825}(13473,\cdot)\)
\(\chi_{26825}(14802,\cdot)\)
\(\chi_{26825}(15527,\cdot)\)
\(\chi_{26825}(15817,\cdot)\)
\(\chi_{26825}(15962,\cdot)\)
\(\chi_{26825}(16228,\cdot)\)
...
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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 };
\((12877,17576,23201)\) → \((e\left(\frac{1}{20}\right),i,e\left(\frac{5}{36}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
| \( \chi_{ 26825 }(13352, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{79}{180}\right)\) | \(e\left(\frac{19}{90}\right)\) | \(e\left(\frac{79}{90}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{25}{36}\right)\) | \(e\left(\frac{19}{60}\right)\) | \(e\left(\frac{19}{45}\right)\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{4}{45}\right)\) | \(e\left(\frac{44}{45}\right)\) |
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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)
