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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1944, base_ring=CyclotomicField(162))
M = H._module
chi = DirichletCharacter(H, M([81,81,143]))
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gp:[g,chi] = znchar(Mod(635, 1944))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1944.635");
| Modulus: | \(1944\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(1944\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(162\) |
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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);
|
\(\chi_{1944}(11,\cdot)\)
\(\chi_{1944}(59,\cdot)\)
\(\chi_{1944}(83,\cdot)\)
\(\chi_{1944}(131,\cdot)\)
\(\chi_{1944}(155,\cdot)\)
\(\chi_{1944}(203,\cdot)\)
\(\chi_{1944}(227,\cdot)\)
\(\chi_{1944}(275,\cdot)\)
\(\chi_{1944}(299,\cdot)\)
\(\chi_{1944}(347,\cdot)\)
\(\chi_{1944}(371,\cdot)\)
\(\chi_{1944}(419,\cdot)\)
\(\chi_{1944}(443,\cdot)\)
\(\chi_{1944}(491,\cdot)\)
\(\chi_{1944}(515,\cdot)\)
\(\chi_{1944}(563,\cdot)\)
\(\chi_{1944}(587,\cdot)\)
\(\chi_{1944}(635,\cdot)\)
\(\chi_{1944}(659,\cdot)\)
\(\chi_{1944}(707,\cdot)\)
\(\chi_{1944}(731,\cdot)\)
\(\chi_{1944}(779,\cdot)\)
\(\chi_{1944}(803,\cdot)\)
\(\chi_{1944}(851,\cdot)\)
\(\chi_{1944}(875,\cdot)\)
\(\chi_{1944}(923,\cdot)\)
\(\chi_{1944}(947,\cdot)\)
\(\chi_{1944}(995,\cdot)\)
\(\chi_{1944}(1019,\cdot)\)
\(\chi_{1944}(1067,\cdot)\)
...
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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 };
\((487,973,1217)\) → \((-1,-1,e\left(\frac{143}{162}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
| \( \chi_{ 1944 }(635, a) \) |
\(1\) | \(1\) | \(e\left(\frac{65}{81}\right)\) | \(e\left(\frac{47}{162}\right)\) | \(e\left(\frac{131}{162}\right)\) | \(e\left(\frac{91}{162}\right)\) | \(e\left(\frac{7}{54}\right)\) | \(e\left(\frac{19}{27}\right)\) | \(e\left(\frac{71}{81}\right)\) | \(e\left(\frac{49}{81}\right)\) | \(e\left(\frac{13}{81}\right)\) | \(e\left(\frac{25}{162}\right)\) |
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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)
