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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(10048, base_ring=CyclotomicField(208))
M = H._module
chi = DirichletCharacter(H, M([104,143,40]))
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gp:[g,chi] = znchar(Mod(2787, 10048))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("10048.2787");
| Modulus: | \(10048\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(10048\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(208\) |
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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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\(\chi_{10048}(27,\cdot)\)
\(\chi_{10048}(275,\cdot)\)
\(\chi_{10048}(363,\cdot)\)
\(\chi_{10048}(475,\cdot)\)
\(\chi_{10048}(739,\cdot)\)
\(\chi_{10048}(771,\cdot)\)
\(\chi_{10048}(843,\cdot)\)
\(\chi_{10048}(867,\cdot)\)
\(\chi_{10048}(875,\cdot)\)
\(\chi_{10048}(1083,\cdot)\)
\(\chi_{10048}(1155,\cdot)\)
\(\chi_{10048}(1163,\cdot)\)
\(\chi_{10048}(1283,\cdot)\)
\(\chi_{10048}(1531,\cdot)\)
\(\chi_{10048}(1619,\cdot)\)
\(\chi_{10048}(1731,\cdot)\)
\(\chi_{10048}(1995,\cdot)\)
\(\chi_{10048}(2027,\cdot)\)
\(\chi_{10048}(2099,\cdot)\)
\(\chi_{10048}(2123,\cdot)\)
\(\chi_{10048}(2131,\cdot)\)
\(\chi_{10048}(2339,\cdot)\)
\(\chi_{10048}(2411,\cdot)\)
\(\chi_{10048}(2419,\cdot)\)
\(\chi_{10048}(2539,\cdot)\)
\(\chi_{10048}(2787,\cdot)\)
\(\chi_{10048}(2875,\cdot)\)
\(\chi_{10048}(2987,\cdot)\)
\(\chi_{10048}(3251,\cdot)\)
\(\chi_{10048}(3283,\cdot)\)
...
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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 };
\((3455,3141,6913)\) → \((-1,e\left(\frac{11}{16}\right),e\left(\frac{5}{26}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 10048 }(2787, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{69}{208}\right)\) | \(e\left(\frac{183}{208}\right)\) | \(e\left(\frac{67}{104}\right)\) | \(e\left(\frac{69}{104}\right)\) | \(e\left(\frac{67}{208}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{11}{52}\right)\) | \(e\left(\frac{49}{52}\right)\) | \(e\left(\frac{33}{208}\right)\) | \(e\left(\frac{203}{208}\right)\) |
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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)
