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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([72,135,28]))
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gp:[g,chi] = znchar(Mod(1247, 3723))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("3723.1247");
| 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: | even |
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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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\(\chi_{3723}(23,\cdot)\)
\(\chi_{3723}(92,\cdot)\)
\(\chi_{3723}(194,\cdot)\)
\(\chi_{3723}(269,\cdot)\)
\(\chi_{3723}(473,\cdot)\)
\(\chi_{3723}(488,\cdot)\)
\(\chi_{3723}(692,\cdot)\)
\(\chi_{3723}(809,\cdot)\)
\(\chi_{3723}(974,\cdot)\)
\(\chi_{3723}(1076,\cdot)\)
\(\chi_{3723}(1133,\cdot)\)
\(\chi_{3723}(1193,\cdot)\)
\(\chi_{3723}(1229,\cdot)\)
\(\chi_{3723}(1247,\cdot)\)
\(\chi_{3723}(1295,\cdot)\)
\(\chi_{3723}(1337,\cdot)\)
\(\chi_{3723}(1448,\cdot)\)
\(\chi_{3723}(1472,\cdot)\)
\(\chi_{3723}(1571,\cdot)\)
\(\chi_{3723}(1625,\cdot)\)
\(\chi_{3723}(1727,\cdot)\)
\(\chi_{3723}(1775,\cdot)\)
\(\chi_{3723}(1850,\cdot)\)
\(\chi_{3723}(1892,\cdot)\)
\(\chi_{3723}(1910,\cdot)\)
\(\chi_{3723}(1952,\cdot)\)
\(\chi_{3723}(2063,\cdot)\)
\(\chi_{3723}(2069,\cdot)\)
\(\chi_{3723}(2105,\cdot)\)
\(\chi_{3723}(2111,\cdot)\)
...
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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 };
\((2483,3505,1684)\) → \((-1,e\left(\frac{15}{16}\right),e\left(\frac{7}{36}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 3723 }(1247, a) \) |
\(1\) | \(1\) | \(e\left(\frac{13}{72}\right)\) | \(e\left(\frac{13}{36}\right)\) | \(e\left(\frac{55}{144}\right)\) | \(e\left(\frac{35}{48}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{109}{144}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{131}{144}\right)\) | \(e\left(\frac{13}{18}\right)\) |
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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)
