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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(55825, base_ring=CyclotomicField(420))
M = H._module
chi = DirichletCharacter(H, M([42,280,84,45]))
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gp:[g,chi] = znchar(Mod(5779, 55825))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("55825.5779");
| Modulus: | \(55825\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(55825\) |
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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: | 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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\(\chi_{55825}(984,\cdot)\)
\(\chi_{55825}(1439,\cdot)\)
\(\chi_{55825}(1719,\cdot)\)
\(\chi_{55825}(1929,\cdot)\)
\(\chi_{55825}(3089,\cdot)\)
\(\chi_{55825}(3644,\cdot)\)
\(\chi_{55825}(3854,\cdot)\)
\(\chi_{55825}(5014,\cdot)\)
\(\chi_{55825}(5289,\cdot)\)
\(\chi_{55825}(5779,\cdot)\)
\(\chi_{55825}(6759,\cdot)\)
\(\chi_{55825}(6939,\cdot)\)
\(\chi_{55825}(7219,\cdot)\)
\(\chi_{55825}(7704,\cdot)\)
\(\chi_{55825}(8864,\cdot)\)
\(\chi_{55825}(10334,\cdot)\)
\(\chi_{55825}(11064,\cdot)\)
\(\chi_{55825}(11279,\cdot)\)
\(\chi_{55825}(12259,\cdot)\)
\(\chi_{55825}(12989,\cdot)\)
\(\chi_{55825}(12994,\cdot)\)
\(\chi_{55825}(14184,\cdot)\)
\(\chi_{55825}(14914,\cdot)\)
\(\chi_{55825}(15194,\cdot)\)
\(\chi_{55825}(16109,\cdot)\)
\(\chi_{55825}(16839,\cdot)\)
\(\chi_{55825}(17054,\cdot)\)
\(\chi_{55825}(18309,\cdot)\)
\(\chi_{55825}(19254,\cdot)\)
\(\chi_{55825}(20234,\cdot)\)
...
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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 };
\((49127,7976,15226,15401)\) → \((e\left(\frac{1}{10}\right),e\left(\frac{2}{3}\right),e\left(\frac{1}{5}\right),e\left(\frac{3}{28}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(8\) | \(9\) | \(12\) | \(13\) | \(16\) | \(17\) |
| \( \chi_{ 55825 }(5779, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{311}{420}\right)\) | \(e\left(\frac{211}{420}\right)\) | \(e\left(\frac{101}{210}\right)\) | \(e\left(\frac{17}{70}\right)\) | \(e\left(\frac{31}{140}\right)\) | \(e\left(\frac{1}{210}\right)\) | \(e\left(\frac{59}{60}\right)\) | \(e\left(\frac{1}{35}\right)\) | \(e\left(\frac{101}{105}\right)\) | \(e\left(\frac{1}{60}\right)\) |
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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)
