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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4257, base_ring=CyclotomicField(210))
M = H._module
chi = DirichletCharacter(H, M([70,42,170]))
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gp:[g,chi] = znchar(Mod(1192, 4257))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("4257.1192");
| Modulus: | \(4257\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(4257\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(105\) |
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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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\(\chi_{4257}(25,\cdot)\)
\(\chi_{4257}(31,\cdot)\)
\(\chi_{4257}(412,\cdot)\)
\(\chi_{4257}(427,\cdot)\)
\(\chi_{4257}(454,\cdot)\)
\(\chi_{4257}(526,\cdot)\)
\(\chi_{4257}(619,\cdot)\)
\(\chi_{4257}(625,\cdot)\)
\(\chi_{4257}(841,\cdot)\)
\(\chi_{4257}(916,\cdot)\)
\(\chi_{4257}(961,\cdot)\)
\(\chi_{4257}(1006,\cdot)\)
\(\chi_{4257}(1186,\cdot)\)
\(\chi_{4257}(1192,\cdot)\)
\(\chi_{4257}(1213,\cdot)\)
\(\chi_{4257}(1303,\cdot)\)
\(\chi_{4257}(1543,\cdot)\)
\(\chi_{4257}(1588,\cdot)\)
\(\chi_{4257}(1600,\cdot)\)
\(\chi_{4257}(1615,\cdot)\)
\(\chi_{4257}(1687,\cdot)\)
\(\chi_{4257}(1780,\cdot)\)
\(\chi_{4257}(1786,\cdot)\)
\(\chi_{4257}(1906,\cdot)\)
\(\chi_{4257}(1930,\cdot)\)
\(\chi_{4257}(2077,\cdot)\)
\(\chi_{4257}(2293,\cdot)\)
\(\chi_{4257}(2347,\cdot)\)
\(\chi_{4257}(2374,\cdot)\)
\(\chi_{4257}(2704,\cdot)\)
...
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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 };
\((947,2323,1981)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{1}{5}\right),e\left(\frac{17}{21}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(13\) | \(14\) | \(16\) | \(17\) |
| \( \chi_{ 4257 }(1192, a) \) |
\(1\) | \(1\) | \(e\left(\frac{41}{105}\right)\) | \(e\left(\frac{82}{105}\right)\) | \(e\left(\frac{74}{105}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{6}{35}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{27}{35}\right)\) | \(e\left(\frac{16}{35}\right)\) | \(e\left(\frac{59}{105}\right)\) | \(e\left(\frac{59}{105}\right)\) |
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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)
