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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7987, base_ring=CyclotomicField(378))
M = H._module
chi = DirichletCharacter(H, M([36,98]))
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gp:[g,chi] = znchar(Mod(1992, 7987))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("7987.1992");
| Modulus: | \(7987\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(7987\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(189\) |
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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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\(\chi_{7987}(65,\cdot)\)
\(\chi_{7987}(135,\cdot)\)
\(\chi_{7987}(158,\cdot)\)
\(\chi_{7987}(184,\cdot)\)
\(\chi_{7987}(240,\cdot)\)
\(\chi_{7987}(452,\cdot)\)
\(\chi_{7987}(550,\cdot)\)
\(\chi_{7987}(674,\cdot)\)
\(\chi_{7987}(767,\cdot)\)
\(\chi_{7987}(788,\cdot)\)
\(\chi_{7987}(807,\cdot)\)
\(\chi_{7987}(821,\cdot)\)
\(\chi_{7987}(879,\cdot)\)
\(\chi_{7987}(1003,\cdot)\)
\(\chi_{7987}(1110,\cdot)\)
\(\chi_{7987}(1276,\cdot)\)
\(\chi_{7987}(1299,\cdot)\)
\(\chi_{7987}(1325,\cdot)\)
\(\chi_{7987}(1381,\cdot)\)
\(\chi_{7987}(1593,\cdot)\)
\(\chi_{7987}(1691,\cdot)\)
\(\chi_{7987}(1815,\cdot)\)
\(\chi_{7987}(1908,\cdot)\)
\(\chi_{7987}(1943,\cdot)\)
\(\chi_{7987}(1948,\cdot)\)
\(\chi_{7987}(1962,\cdot)\)
\(\chi_{7987}(1992,\cdot)\)
\(\chi_{7987}(2020,\cdot)\)
\(\chi_{7987}(2102,\cdot)\)
\(\chi_{7987}(2144,\cdot)\)
...
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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 };
\((7010,491)\) → \((e\left(\frac{2}{21}\right),e\left(\frac{7}{27}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
| \( \chi_{ 7987 }(1992, a) \) |
\(1\) | \(1\) | \(e\left(\frac{139}{189}\right)\) | \(e\left(\frac{53}{189}\right)\) | \(e\left(\frac{89}{189}\right)\) | \(e\left(\frac{41}{63}\right)\) | \(e\left(\frac{1}{63}\right)\) | \(e\left(\frac{13}{63}\right)\) | \(e\left(\frac{106}{189}\right)\) | \(e\left(\frac{73}{189}\right)\) | \(e\left(\frac{188}{189}\right)\) | \(e\left(\frac{142}{189}\right)\) |
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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)
