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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8388, base_ring=CyclotomicField(348))
M = H._module
chi = DirichletCharacter(H, M([174,58,21]))
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gp:[g,chi] = znchar(Mod(1343, 8388))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("8388.1343");
| Modulus: | \(8388\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(8388\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(348\) |
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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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\(\chi_{8388}(167,\cdot)\)
\(\chi_{8388}(203,\cdot)\)
\(\chi_{8388}(263,\cdot)\)
\(\chi_{8388}(299,\cdot)\)
\(\chi_{8388}(479,\cdot)\)
\(\chi_{8388}(491,\cdot)\)
\(\chi_{8388}(587,\cdot)\)
\(\chi_{8388}(599,\cdot)\)
\(\chi_{8388}(671,\cdot)\)
\(\chi_{8388}(803,\cdot)\)
\(\chi_{8388}(923,\cdot)\)
\(\chi_{8388}(947,\cdot)\)
\(\chi_{8388}(1055,\cdot)\)
\(\chi_{8388}(1103,\cdot)\)
\(\chi_{8388}(1139,\cdot)\)
\(\chi_{8388}(1343,\cdot)\)
\(\chi_{8388}(1391,\cdot)\)
\(\chi_{8388}(1499,\cdot)\)
\(\chi_{8388}(1559,\cdot)\)
\(\chi_{8388}(1571,\cdot)\)
\(\chi_{8388}(1595,\cdot)\)
\(\chi_{8388}(1667,\cdot)\)
\(\chi_{8388}(1703,\cdot)\)
\(\chi_{8388}(1751,\cdot)\)
\(\chi_{8388}(1895,\cdot)\)
\(\chi_{8388}(1919,\cdot)\)
\(\chi_{8388}(2111,\cdot)\)
\(\chi_{8388}(2147,\cdot)\)
\(\chi_{8388}(2207,\cdot)\)
\(\chi_{8388}(2315,\cdot)\)
...
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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 };
\((4195,1865,469)\) → \((-1,e\left(\frac{1}{6}\right),e\left(\frac{7}{116}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
| \( \chi_{ 8388 }(1343, a) \) |
\(1\) | \(1\) | \(e\left(\frac{275}{348}\right)\) | \(e\left(\frac{49}{87}\right)\) | \(e\left(\frac{193}{348}\right)\) | \(e\left(\frac{151}{174}\right)\) | \(e\left(\frac{83}{116}\right)\) | \(e\left(\frac{41}{58}\right)\) | \(e\left(\frac{8}{87}\right)\) | \(e\left(\frac{101}{174}\right)\) | \(e\left(\frac{23}{174}\right)\) | \(e\left(\frac{71}{87}\right)\) |
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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)
