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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9408, base_ring=CyclotomicField(336))
M = H._module
chi = DirichletCharacter(H, M([168,21,168,104]))
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gp:[g,chi] = znchar(Mod(59, 9408))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("9408.59");
| Modulus: | \(9408\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(9408\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(336\) |
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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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\(\chi_{9408}(59,\cdot)\)
\(\chi_{9408}(131,\cdot)\)
\(\chi_{9408}(299,\cdot)\)
\(\chi_{9408}(395,\cdot)\)
\(\chi_{9408}(467,\cdot)\)
\(\chi_{9408}(563,\cdot)\)
\(\chi_{9408}(635,\cdot)\)
\(\chi_{9408}(731,\cdot)\)
\(\chi_{9408}(899,\cdot)\)
\(\chi_{9408}(971,\cdot)\)
\(\chi_{9408}(1067,\cdot)\)
\(\chi_{9408}(1139,\cdot)\)
\(\chi_{9408}(1235,\cdot)\)
\(\chi_{9408}(1307,\cdot)\)
\(\chi_{9408}(1475,\cdot)\)
\(\chi_{9408}(1571,\cdot)\)
\(\chi_{9408}(1643,\cdot)\)
\(\chi_{9408}(1739,\cdot)\)
\(\chi_{9408}(1811,\cdot)\)
\(\chi_{9408}(1907,\cdot)\)
\(\chi_{9408}(2075,\cdot)\)
\(\chi_{9408}(2147,\cdot)\)
\(\chi_{9408}(2243,\cdot)\)
\(\chi_{9408}(2315,\cdot)\)
\(\chi_{9408}(2411,\cdot)\)
\(\chi_{9408}(2483,\cdot)\)
\(\chi_{9408}(2651,\cdot)\)
\(\chi_{9408}(2747,\cdot)\)
\(\chi_{9408}(2819,\cdot)\)
\(\chi_{9408}(2915,\cdot)\)
...
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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 };
\((1471,6469,3137,4609)\) → \((-1,e\left(\frac{1}{16}\right),-1,e\left(\frac{13}{42}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) |
| \( \chi_{ 9408 }(59, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{181}{336}\right)\) | \(e\left(\frac{233}{336}\right)\) | \(e\left(\frac{17}{112}\right)\) | \(e\left(\frac{83}{84}\right)\) | \(e\left(\frac{37}{48}\right)\) | \(e\left(\frac{107}{168}\right)\) | \(e\left(\frac{13}{168}\right)\) | \(e\left(\frac{85}{112}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{157}{336}\right)\) |
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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)
