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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3783, base_ring=CyclotomicField(96))
M = H._module
chi = DirichletCharacter(H, M([48,24,85]))
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gp:[g,chi] = znchar(Mod(2660, 3783))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("3783.2660");
| Modulus: | \(3783\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(3783\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(96\) |
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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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\(\chi_{3783}(317,\cdot)\)
\(\chi_{3783}(320,\cdot)\)
\(\chi_{3783}(359,\cdot)\)
\(\chi_{3783}(395,\cdot)\)
\(\chi_{3783}(863,\cdot)\)
\(\chi_{3783}(980,\cdot)\)
\(\chi_{3783}(983,\cdot)\)
\(\chi_{3783}(1178,\cdot)\)
\(\chi_{3783}(1256,\cdot)\)
\(\chi_{3783}(1448,\cdot)\)
\(\chi_{3783}(1526,\cdot)\)
\(\chi_{3783}(1529,\cdot)\)
\(\chi_{3783}(1763,\cdot)\)
\(\chi_{3783}(1802,\cdot)\)
\(\chi_{3783}(1919,\cdot)\)
\(\chi_{3783}(1955,\cdot)\)
\(\chi_{3783}(2462,\cdot)\)
\(\chi_{3783}(2543,\cdot)\)
\(\chi_{3783}(2579,\cdot)\)
\(\chi_{3783}(2657,\cdot)\)
\(\chi_{3783}(2660,\cdot)\)
\(\chi_{3783}(2699,\cdot)\)
\(\chi_{3783}(2774,\cdot)\)
\(\chi_{3783}(2852,\cdot)\)
\(\chi_{3783}(2933,\cdot)\)
\(\chi_{3783}(2969,\cdot)\)
\(\chi_{3783}(3047,\cdot)\)
\(\chi_{3783}(3164,\cdot)\)
\(\chi_{3783}(3206,\cdot)\)
\(\chi_{3783}(3284,\cdot)\)
...
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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 };
\((1262,2329,781)\) → \((-1,i,e\left(\frac{85}{96}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(14\) | \(16\) | \(17\) |
| \( \chi_{ 3783 }(2660, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{41}{48}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{61}{96}\right)\) | \(e\left(\frac{19}{96}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{47}{96}\right)\) | \(e\left(\frac{19}{48}\right)\) | \(e\left(\frac{5}{96}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{77}{96}\right)\) |
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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)
