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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(10944, base_ring=CyclotomicField(144))
M = H._module
chi = DirichletCharacter(H, M([0,63,24,40]))
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gp:[g,chi] = znchar(Mod(3053, 10944))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("10944.3053");
| Modulus: | \(10944\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(10944\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(144\) |
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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: | even |
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sage:chi.is_odd()
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gp:zncharisodd(g,chi)
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magma:IsOdd(chi);
|
\(\chi_{10944}(29,\cdot)\)
\(\chi_{10944}(317,\cdot)\)
\(\chi_{10944}(725,\cdot)\)
\(\chi_{10944}(869,\cdot)\)
\(\chi_{10944}(965,\cdot)\)
\(\chi_{10944}(1085,\cdot)\)
\(\chi_{10944}(1397,\cdot)\)
\(\chi_{10944}(1685,\cdot)\)
\(\chi_{10944}(2093,\cdot)\)
\(\chi_{10944}(2237,\cdot)\)
\(\chi_{10944}(2333,\cdot)\)
\(\chi_{10944}(2453,\cdot)\)
\(\chi_{10944}(2765,\cdot)\)
\(\chi_{10944}(3053,\cdot)\)
\(\chi_{10944}(3461,\cdot)\)
\(\chi_{10944}(3605,\cdot)\)
\(\chi_{10944}(3701,\cdot)\)
\(\chi_{10944}(3821,\cdot)\)
\(\chi_{10944}(4133,\cdot)\)
\(\chi_{10944}(4421,\cdot)\)
\(\chi_{10944}(4829,\cdot)\)
\(\chi_{10944}(4973,\cdot)\)
\(\chi_{10944}(5069,\cdot)\)
\(\chi_{10944}(5189,\cdot)\)
\(\chi_{10944}(5501,\cdot)\)
\(\chi_{10944}(5789,\cdot)\)
\(\chi_{10944}(6197,\cdot)\)
\(\chi_{10944}(6341,\cdot)\)
\(\chi_{10944}(6437,\cdot)\)
\(\chi_{10944}(6557,\cdot)\)
...
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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 };
\((9919,2053,1217,9217)\) → \((1,e\left(\frac{7}{16}\right),e\left(\frac{1}{6}\right),e\left(\frac{5}{18}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 10944 }(3053, a) \) |
\(1\) | \(1\) | \(e\left(\frac{103}{144}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{41}{144}\right)\) | \(e\left(\frac{19}{36}\right)\) | \(e\left(\frac{37}{72}\right)\) | \(e\left(\frac{31}{72}\right)\) | \(e\left(\frac{101}{144}\right)\) | \(1\) | \(e\left(\frac{61}{144}\right)\) |
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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)
