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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(220669, base_ring=CyclotomicField(740))
M = H._module
chi = DirichletCharacter(H, M([175,699]))
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gp:[g,chi] = znchar(Mod(5055, 220669))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("220669.5055");
| Modulus: | \(220669\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(220669\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(740\) |
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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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\(\chi_{220669}(644,\cdot)\)
\(\chi_{220669}(1356,\cdot)\)
\(\chi_{220669}(1740,\cdot)\)
\(\chi_{220669}(1777,\cdot)\)
\(\chi_{220669}(2512,\cdot)\)
\(\chi_{220669}(2693,\cdot)\)
\(\chi_{220669}(3288,\cdot)\)
\(\chi_{220669}(5055,\cdot)\)
\(\chi_{220669}(5326,\cdot)\)
\(\chi_{220669}(5545,\cdot)\)
\(\chi_{220669}(7037,\cdot)\)
\(\chi_{220669}(7044,\cdot)\)
\(\chi_{220669}(7073,\cdot)\)
\(\chi_{220669}(7086,\cdot)\)
\(\chi_{220669}(7311,\cdot)\)
\(\chi_{220669}(11029,\cdot)\)
\(\chi_{220669}(11044,\cdot)\)
\(\chi_{220669}(11259,\cdot)\)
\(\chi_{220669}(11531,\cdot)\)
\(\chi_{220669}(11678,\cdot)\)
\(\chi_{220669}(11992,\cdot)\)
\(\chi_{220669}(12090,\cdot)\)
\(\chi_{220669}(12677,\cdot)\)
\(\chi_{220669}(13015,\cdot)\)
\(\chi_{220669}(13392,\cdot)\)
\(\chi_{220669}(13720,\cdot)\)
\(\chi_{220669}(13880,\cdot)\)
\(\chi_{220669}(14291,\cdot)\)
\(\chi_{220669}(17752,\cdot)\)
\(\chi_{220669}(17937,\cdot)\)
...
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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 };
\((48874,122926)\) → \((e\left(\frac{35}{148}\right),e\left(\frac{699}{740}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 220669 }(5055, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{727}{740}\right)\) | \(e\left(\frac{96}{185}\right)\) | \(e\left(\frac{357}{370}\right)\) | \(e\left(\frac{241}{370}\right)\) | \(e\left(\frac{371}{740}\right)\) | \(e\left(\frac{3}{370}\right)\) | \(e\left(\frac{701}{740}\right)\) | \(e\left(\frac{7}{185}\right)\) | \(e\left(\frac{469}{740}\right)\) | \(e\left(\frac{14}{37}\right)\) |
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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)
