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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3723, base_ring=CyclotomicField(144))
M = H._module
chi = DirichletCharacter(H, M([72,99,74]))
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gp:[g,chi] = znchar(Mod(1163, 3723))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("3723.1163");
| Modulus: | \(3723\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(3723\) |
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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: | 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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\(\chi_{3723}(29,\cdot)\)
\(\chi_{3723}(113,\cdot)\)
\(\chi_{3723}(131,\cdot)\)
\(\chi_{3723}(398,\cdot)\)
\(\chi_{3723}(464,\cdot)\)
\(\chi_{3723}(623,\cdot)\)
\(\chi_{3723}(626,\cdot)\)
\(\chi_{3723}(719,\cdot)\)
\(\chi_{3723}(743,\cdot)\)
\(\chi_{3723}(887,\cdot)\)
\(\chi_{3723}(890,\cdot)\)
\(\chi_{3723}(1142,\cdot)\)
\(\chi_{3723}(1163,\cdot)\)
\(\chi_{3723}(1196,\cdot)\)
\(\chi_{3723}(1367,\cdot)\)
\(\chi_{3723}(1421,\cdot)\)
\(\chi_{3723}(1544,\cdot)\)
\(\chi_{3723}(1553,\cdot)\)
\(\chi_{3723}(1559,\cdot)\)
\(\chi_{3723}(1637,\cdot)\)
\(\chi_{3723}(1724,\cdot)\)
\(\chi_{3723}(1757,\cdot)\)
\(\chi_{3723}(1799,\cdot)\)
\(\chi_{3723}(1814,\cdot)\)
\(\chi_{3723}(1931,\cdot)\)
\(\chi_{3723}(1958,\cdot)\)
\(\chi_{3723}(2030,\cdot)\)
\(\chi_{3723}(2204,\cdot)\)
\(\chi_{3723}(2510,\cdot)\)
\(\chi_{3723}(2522,\cdot)\)
...
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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 };
\((2483,3505,1684)\) → \((-1,e\left(\frac{11}{16}\right),e\left(\frac{37}{72}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 3723 }(1163, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{17}{72}\right)\) | \(e\left(\frac{17}{36}\right)\) | \(e\left(\frac{65}{144}\right)\) | \(e\left(\frac{25}{48}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{83}{144}\right)\) | \(e\left(\frac{5}{72}\right)\) | \(e\left(\frac{109}{144}\right)\) | \(e\left(\frac{17}{18}\right)\) |
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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)
