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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(11825, base_ring=CyclotomicField(420))
M = H._module
chi = DirichletCharacter(H, M([105,168,130]))
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gp:[g,chi] = znchar(Mod(1732, 11825))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("11825.1732");
| Modulus: | \(11825\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(2365\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(420\) |
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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 \(\chi_{2365}(1732,\cdot)\) |
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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);
|
\(\chi_{11825}(157,\cdot)\)
\(\chi_{11825}(218,\cdot)\)
\(\chi_{11825}(493,\cdot)\)
\(\chi_{11825}(632,\cdot)\)
\(\chi_{11825}(707,\cdot)\)
\(\chi_{11825}(718,\cdot)\)
\(\chi_{11825}(757,\cdot)\)
\(\chi_{11825}(807,\cdot)\)
\(\chi_{11825}(1093,\cdot)\)
\(\chi_{11825}(1318,\cdot)\)
\(\chi_{11825}(1582,\cdot)\)
\(\chi_{11825}(1732,\cdot)\)
\(\chi_{11825}(1818,\cdot)\)
\(\chi_{11825}(1868,\cdot)\)
\(\chi_{11825}(1918,\cdot)\)
\(\chi_{11825}(2007,\cdot)\)
\(\chi_{11825}(2082,\cdot)\)
\(\chi_{11825}(2093,\cdot)\)
\(\chi_{11825}(2282,\cdot)\)
\(\chi_{11825}(2368,\cdot)\)
\(\chi_{11825}(2557,\cdot)\)
\(\chi_{11825}(2643,\cdot)\)
\(\chi_{11825}(2743,\cdot)\)
\(\chi_{11825}(2843,\cdot)\)
\(\chi_{11825}(2907,\cdot)\)
\(\chi_{11825}(2957,\cdot)\)
\(\chi_{11825}(3243,\cdot)\)
\(\chi_{11825}(3382,\cdot)\)
\(\chi_{11825}(3468,\cdot)\)
\(\chi_{11825}(3732,\cdot)\)
...
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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 };
\((5677,7526,9076)\) → \((i,e\left(\frac{2}{5}\right),e\left(\frac{13}{42}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(12\) | \(13\) | \(14\) |
| \( \chi_{ 11825 }(1732, a) \) |
\(1\) | \(1\) | \(e\left(\frac{1}{140}\right)\) | \(e\left(\frac{109}{420}\right)\) | \(e\left(\frac{1}{70}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{3}{140}\right)\) | \(e\left(\frac{109}{210}\right)\) | \(e\left(\frac{23}{84}\right)\) | \(e\left(\frac{23}{420}\right)\) | \(e\left(\frac{187}{210}\right)\) |
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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)
